Torcath Lpereary é, (Peat E lt Digitized by the Internet Archive s in 2010 with funding from University of Toronto ~ " =<» oe ht tp:/www.archive.org/details/fo restmensuratio00grav - - oe. ” o> V7. + \ ve! cc) 1 wes ve ee <8): ~ FOREST MENSURATION BY HENRY SOLON GRAVES, M.A. Director of the Forest School, Yale Untverstly FIRST EDITION FIRST THOUSAND NEW YORK A eee JOHN WILEY & SONS \ Asan Te Lonpon: CHAPMAN & HALL, LIMITED 1906 G8 Cen. +s 3 Copyright, 1906 ; BY ; HENRY SOLON GRAVES ROBERT DRUMMOND, PRINTER, NEW YORK TO MY FRIEND George Dudley Seymour RREFACE, Tue urgent need of a reference book in forest mensuration for class work at the Yale Forest School has induced me to publish my lectures, given during the past year, with such addi- tions as are necessary to present the material in the form of a book. This book is designed as a guide for students of forestry and as a reference book for practical foresters and lumbermen. It is not intended that it should replace the field instruction of the student of forest mensuration, for the subject cannot be mastered except through training and experience in the woods. Not only is it impossible to acquire from books the ability to estimate timber and to scale logs, but the scientific work in forest mensuration equally requires field experience. On the other hand, a systematic work describing the principles of forest measurements is of great service to a teacher in con- ducting his field instruction, and such a book should be of constant use to a forester, especially if he is engaged in research work. The material for the book has been selected with these principles in mind, and I have endeavored to furnish information which will be useful both to practical business men and to those engaged in work of investigation. I have drawn freely from the experience of European foresters, with particular reference to their methods of measuring logs, determining the age of stands, and studying growth. 4 vi PREFACE, I have made special use of the works of Dr. Franz Baur and Dr. Udo Miller. Acknowledgments are due to Mr. H. D. Tiemann for his assistance in preparing Chapters V and XVIII, to Mr. J. O. Hopwood for compiling the laws relating to the measurement of logs, and to Mr. Raphael Zon and Mr. Austin W. Hawes for reviewing certain portions of the work. HENRY S. GRAVES. YALE UNIVERSITY, New Haven, 'Conn., August 1, 1906. SEC. ah WwW ND CONTENTS. CHAPTER. I, INTRODUCTION, PAGE Definition of Forest Mensuration............0. SO ree ee I Importance of the Study of Forest Mensuration.............. 2 ROUAR ER AR HEME Cts lcav' 3s steve aijs. o- Loasole ed a ENT a ts ea aN Rygine s «Batten ae 7 RIMES GE WIG ASUTOUGICN Us. -ds, oc sidieen Sica eees er ee oe 8 CHAPTER II. DETERMINATION OF THE CONTENTS OF Locs IN BoarpD MEASURE. Dennition. Of Board, Measure, i. sence as code oon-auieeeo.. oe II Board Measure Applied to Round Logs... oo. coe v ene 12 Principles of Constructing: Log (Rales: svi. ii aa 12 . Conditions Influencing the Contents of Logs in Board Feet.... 13 . Value of Board Measure Applied to Round Logs............ 14 . Need of Uniformity in the Measurement of Logs............ ay] . Principles of Constructing a Standard Log Rule.............. 18 Pmeieesion Of A Standard Ieteln) 5 25 <,h cid s sistas occ.¢ 6 3's ee 0:ou oes 20 Reena tuciion Of Local Loo Rales so). sis. dese oho ce wae dele San 21 ents LAI OS. Ice. ken vk heave vip ioen ee ws ptt Riki w Ninety aa «Sheeran mameo miles panctioned DY Statute; .. ow siws tee sie auc vie becdiem 21 . Which is the Best Log Rule in Common Use?............... 22 any tasetory.of Log Rules. .. 1 ceccee esse Pereye sn ee ee 22 emameatana OL WO RWC... < ijn'e's. 06 6 00 vive vibi6 v0.00 6 010 s/s vinbisn 26 viii CONTENTS. CHAPTER III. DETAILED Discussion oF LoG RULEs, SEC. PAGE 19. The Champlain Rule......--eeeeer eres esse seer eceeencees 27 so. Daniels’ Universal Log Rule... . 1. eee cece eter cere eee eees 2 2. The International Log Rule... .. 1... eee e cece reece renee eens 35 222, The Doyle Rule a nc lee deemiel sls eielata es (stave es Seale eel St ouet ere ee ate n ee 38 23. The Copier FG 4 csi rb « wiesne/e state's & oceln On cals auniesace ctaamin ons 4! 24. The Maine [Tt SN NOR Er PNP ain REE ay ce Ge (RAPER 2 2s. The Hanna Rule.......esee ee reeeee cere ereeceer erences 45 26. The Spaulding Rule... 1... eee ee eect ee erence ee eee eee 45 27. The British Cicsterrns i RISD ats es es 0 ots is a ac aieie se wee Smee 46 BS) The Drew. Ratios. « cse-to/e ations ousis, #'0n, 5 0 eie\n acs eiprote sta inie h co (wieght 46 29. The Constantine Rule... .. + sees eee e ee eee eee e eee e eee e eens 47 30. The Canadian Rules.....--.s+sesereeeer ee eee ee eeeseerees 47 31. Miscellaneous Log Rules... .--.+++eeeee eer cece eee ee cece: 48 CHAPTER IV. Loc RuLes BASED ON STANDARDS. 32. Definition of Stamelard Meastire. soc aix ic) suis ow eewlace re s'G a Mean nen 53 33. The Nineteen-inch Standard Gey ts siakvs a's thos. 5 se mlgraeee 54 34. The New Hampshire Rule (Blodvett: Rule)}s..:.;. sos ssh ines s oe 55 35. The Cube Rule... 1... ces cece eet e esc e ces tence eeeeecens 57 36. Other Standard Rules......-..-+sseeeeee ee eeee tere eeeeees e7 CHAPTER V. Metuops oF SCALING LOGs. 37. Instruments for Scaling TO chills oa ly dea lates mile @ atc ie eee ee 60 38. Methods of Measuring the Diameters and Lengths........... 62 39. Methods of Making Discount for elects 5s An eae terse ete Moe 40. Rules for Scaling Used on the Forest Reserves........+.+- ES Cake CHAPTER VI. DETERMINATION OF THE CONTENTS OF LOGs IN CuBic FEET. 41. Use of the Cubic Foot in America.....----+ss-srsresteees 76 42. The Measurement of Logs to Determine their Cubic Contents.. 77 43. Measuring Instruments... 005g 0's ne Neen es a a 79 44. Principles Underlying the Determination of the Cubic Contents of Logs ane Trees fn. e000 oC sp ae eas pedo ne 85 49. 50. 55: 56. o/* 58. 59. 60. Ox. 62. 63. 64. CONTENTS. PEA ASTI eRe TT CPTI TILES 0s vi otelntc Foul a c's ou © bc-ole ws 0 be elale Beals * eterna wom or Sectional ATeGas.. isis ociek cade nabedee¥ee . Methods of Determining the Volume of Logs by the Measure- ment of the End Diameters and the Length............. 8. Method of Cubing Logs by the Measurement of the Length and of the Diameter at. the Middle... 2.0.0. ces eens ba os Method of Cubing Logs by the Measurement of the Length, the Top Diameter, and the Diameter at One-third the AS EMCICs Intoriee Cale AMI UT UL cinch sake be get, wiac debe w ei) $s oes espn Method of Cubing Logs by the Measurement of Two End Diameters, the Middle Diameter, and the Length of the IS Me UIOG COT PINtih. Cae eli stay peas. iv hie ak a Ss she oe teers ly, 3 ere soOuarter-cittis MepMOy te yh etay Bl cua wie quis Shae do's, 0 . Other Methods Chiefly of Scientific Interest.............e00. aoontents of-Logs, moCu bie Meiers: sis cmssinbi e's soe w e.4e 6 oie CHAPTER VII. DETERMINATION OF THE CUBIC CONTENTS OF SQUARED LoGs. Method of ormcer \Girtli ous ne ote arena cs anv ake bore perenne sy Aiba Pi wOsemana st INO ys 1. Us sees eae ee som eed et awee) waa 5: Wears he stnschibeavaquare Rule... 2255 fo--scpeh ero aae ted Dates Sas . CHAPTER Viti, Corp MEASURE. The, Measurement. of Cord-wood) 1. Fans sss ve dk A eee Amount.0f Solid’ Wood. tinea Stacked Cord. 3s 2 oe ed Be Relation between Cord Measure and Board and Standard BPCASUICES. |. 55 ss eet ie MES tye Ft wh I aie sah SS wa a OE haat ; CHAPTER: 1X. THE CONTENTS OF ENTIRE FELLED TREES. The Measurement of Entire Felled Trees mame oorupuEation of Volume... 3)! Fos. le pen eels, The Measurement of Crown, Clear Length, and Merchantable eMC PRONE CY ie near Sella ava fone! avy tere exw eie & Lehane armas 94 95 95 96 97 98 99 99 100 LEE II4 x CONTENTS. CHAPTER X. Tue DETERMINATION OF THE HEIGHT OF STANDING TREES. SEC. PAGE 65. Rough Methods of Measurement,.......ssereeeeecveeecrereces 120 GE ion, Measures... 44 ates. cones © ane ae aak ROraeny aricie dik © p> 122 67. The Faustmann Height Measure........:sceccsessncassess 122 63. The Weise Floisitt. Meastres 5.5 cits ence kee ie a ae Pa 129 Go, The Christeri Height. Meastire...:. sj. + 0s eda alvin «dean vein 131 wo, The Klaussner Height Measure...) . 1.1 us wvmansse es v's ss 133 71. The Winkler Height Measure and Dendrometer............. 137 9. The Brandis Heieht Measure... oi. . «0's. oitatan seed ae I4I 73. Clinometer for Measuring Heights............eeseeeeeeeees 144 74. The Abney Hand Level and Clinometer.................... 145. ye. Other Height Measures. .... 0... 0.500 - sees es cere ner atn nsec 147 76. General Directions for the Measurement of Heights of Trees .. 148. 77. Choice of a Height Measure...... 2... cece cere reece cecenes 149 78. The Use of Dendrometers...........eeeeeereecececcccccens 150 CHAPTER XI. DETERMINATION OF THE CONTENTS OF STANDING TREES. ea mratamadte’ DY The Tye. 0 ie i vacate ow alsin aoee a's Sipe ee 152 80. Estimate of the Contents of Standing Trees by Volume Tables end. Porm: Factors .) ss. .i% aats pieced wie kansas eee Tee 81. Rough Method of Estimating the Cubic Contents of Standing TE FOGS ba Fis aia hie a bis wom Sean tl ge ae wa ok olin eee help lel 155 Ba. rrossteldt’s Method «a: scsi: a's. 0:ss = nie ne sis hos! o wig min Wiel is inion 155 $3; Pressler's Method. .<.0.ccin ow cs Lniaie aisle sine estes Svaceroht inlevade’s Gees 155. CHAPTER XII. VOLUME TABLES, 84. Definition of Volume Tables<.) i. Fs. Sees a A ee 158 8s. Volume Tables for Trees of Different Diameters............. 159 86. Volume Tables for Trees Grouped by Diameter and Number OF TGS oo in we lacs o Seep) Denna e Le tg ee ee ws ek 163 87. Volume Tables for Trees of Different Diameters and Merchant- eubyles Spam reas 5 5 tt ah ee re es Ses 164 88. Volume Tables for Trees of Different Diameters and Tree Classes. 6 £2) ie So. 6 Dace ee A ae Ree ae wd cose 164 CONTENTS. SEC 89. Volume Tables for Trees of Different Diameters and Heights. . Cin: Com emRS + LONOB yi 0e e:-acse nave. ssoyesen owe caruovvosw.iove webloie co iba dl CHAPTER XIII. ForM Factors, Rou Gn Or. Mort Pastors: sts. oe ae dc Mee em Pee laser On i Ort LP ACTOLSY, 24.05 ne! oi" age es Ch 93. Variations in the Value of Breast-height Form Factors....... g4. Construction of Tables‘of Form Factors.:...0......0)...... Beceereeutiues Le OM CLO cr. fe wlae’s wield abakagtoaetin | Lin Gace ele aoe Pern Oca ae OI NACEOL migratetesaret ss cee eee cud. oc ee, g7. ihe Conception and Use ofiorm Exponents.........20).2.., eM eR OL ACLOL Ua sy... «keen hen. sae. en Beebe 99. Special: Methods of Determining Form Factors....v......... Pea. Mento. Oivoertn Quotientss ee, tiaras su nel A or. litte cte diene: CHAPTER: XivV. DETERMINATION OF THE CONTENTS OF STANDS. 1o1. Problems of Determining the Contents of Stands............ Seer ter Chaitin. ss. 25k Sx ws See eters ter eee fw et ae etry eee SUIttIL Ge DY JUMGUED VO... s,s cad ac SE aug eters ne ME Cee Rie ee 104. Estimate by the Inspection of Each Tree ina Stand.......... aon owe LOmmU sed in: Mickierar. J re Oe on fg Fe ls tu shh 106. Estimating by Working over Small Squares................. Soyer eseminatine in 4o-1od. Stips-Oaas sath steccye kr eh. os Boe OL ove thy tol rasta cad ey 5 128 MA ea 2 zoo. Estimate: by Wse-at Standahables fre och. ooo alk. oe one ee oe io. motimiate by the Use olmaluation Surveyg: .........e.. <2. rag ete Se Ot Strip: Surge so oo oa os ea wa x oad ne cle bcc ud mee SIMSRINDUtiOn Of the SiimeeeUrVeyS © ooo. ce ace d ls chek a stele: ae te GI EP OTESE Mae. heel eS hav ated ony eee kok ba ee ees mere rvedcitrement :OL the Wiese sips. cee lati ene ete i is eek. Peecrormine The Measuren@emtserny. 2 0 in nafs es So viede acc eee dale we 116. Number of Strip Surveys Required for an Estimate.......... Reerantage. Of otrip SUrVEYS). oo li. fe eae welds cee ul mania” Diothe ot rVveys on iy os fl pike wcoecdle cs tives ot ove bw ea hnd” 119. Instruments Used in Laying Off Sample Plots............... 120. Necessary Precision in Laying Off Sample Plots............. Pour ead ize Ol Sample “Plots: ... 65. 6. ee. a ie'cs op ae dea xi PAGE 166 169 174 176 176 179 182 182 182 184 185 188 Xli CONTENTS. SEC , PAGE 122. Marking the Boundaries of Sample Plots.......... re 215 PEE, COG POPING, 0 5» c's win's 6s ¢ tou xt) Rgwrone wills wep 6.0's:5 Tete 216 ra, The Location of Sample.Plots.. joi... vs si = s'> beets eg eee 216 125. Computation of Volume of the Trees on Valuation Areas..... 219 126. Determination of Volume of Stands by the Use of Felled satmiple Trees stay vac pels pitts 6 huite' c's >.c « 0s peg ea 224 197. The Mean Sample-tree Method. «.....c0.ceseccevnceununtie 224 ras. Arbitrary Group Metmbod. 20 ih6 esc cakes ce 0 0 00 gee 22 rao. The Volume-curve Method)... 6 .icis acca css 0 ae sca stele ee 232 a0, The’ Drandt Method . eis .c ais tke ux va A Ss 0 0 0 eb 233 zor. The Urich Method. ec... sacilep wipes aioe «3.0» « vp gee 236 nad. Hartig's: Methods: yon 0's Sled eee uh Ae eo» ee 240 no2; Blook's Method. «!is....5\. 1 as stecto eset HOE eta wala 0 ox oak 240 r34.' Method of Forest.Form Faetor...\.<2te diay sees sss. ee 241 ran.. Metzger's Method «., « sccm 258 146. Instruments for Measuring Diameter Growth............... 260 147. Counting the Annual Rings from the Pith Outward.......... 261 148. Counting the Rings from Bark toward Pith............ aye 262 149. Investigations of Diameter Groowe. icaiccoescus s,s. akc 2605 150. Determination of Diameter Growtliiccs lenie sc «. . +:0cee ee 266 151. Separate Studies Made for Trees Growing under Different Con- GitiONS os. a ce SRD EAOS vee. sen 268 152. The Study of Diameter Growth in Even-aged Stands......... 268 CONTENTS. xiii SEC, PAGE 153. The Study of Maximum Diameter Growth in Even-aged MRS Rpt xian’: dNaig'c iw Am eastusghe arty SUG Baie weet eu Y 272 154. The Measurement of the Minimum Diameter Growth........ 273 ihe. Stitmuarea erowthratter. Uhinning. 0) ev te Wille soa ae 274 156. Rate of Growth in Uneven-aged Stands.........00.00....... 2765 157. Determination of the Mean Annual Growth of Trees in Uneven- OLAS Es [SESSA SOON Ae ue eA ee RO A co 276 Peep recicuion of Growth for Short’ Periods) oi 0... vdeo. 27 159. Tables of Growth of Trees of Different Diameters............ 278 meaty of the: Growth of Trees Area) 80) elie J. ded os. 282 fae the Growth in Height of Individual Trees:....) 203i. 7.5... 283 162. Determination of the Rate of Growth of Trees in Height.... 284 mee incisit: Growth of Even-aged Stands: 0.0. py mele ane. 287 164. The Growth in Height of Trees of Different Diameters........ 288 165. Study of the Rate of Growth of Individual Trees in Volume.. 290 166. Graphic Method of Determining the Average Volume Growth Gra: Group; ot Trees? i wee pae. et he a a ee 293 167, Modification of the Method by the’ Author.....:.......2..04 295 168. Determination of Volume Growth for Short Periods........: 298 gees tere Gie NVOGWICALION. . ... violation of contracts to cut timber to a specified diameter. Thus contracts are often made in the spruce sections for the cutting of trees above 1o or 12 inches in diameter. Frequently these contracts are broken by cutting all trees down to 6 or 8 inches in diameter. The owner of the land, according to the present custom, is able to collect as damages no more than the stumpage value of the trees wrongfully cut, and he receives nothing what- ever for the injury to the producing power of his forest, although such wrongful cutting may have reduced the growth of the desired timber on his land 50 per cent. If the contract had been faith- fully performed, his timber might have grown, after lumbering, at the rate of 100 board feet per acre per annum, whereas the growth has been reduced by wrongful cutting to 50, or even less, board feet per annum. The real damage to the land in such a case can be reached only through a knowledge of growth. To meet these demands foresters will specialize and become recognized as experts in determining the prospective value of timber lands, and as such will be called on to testify in court in damage cases and also consulted by would-be investors in timber lands. 3. Literature.—English literature on forest mensuration is very meagre. Several works on general forestry and on forest management devote a few chapters to the subject, as, for example, Schlich’s Manual of Forestry, MacGregor’s Organization and Valuation of Forests, The Forester, by Brown and Nisbet, and Green’s Principles of American Forestry ; but none of these works treat forest mensuration fully. By far the best discussion of the subject is contained in the third volume of Schlich’s Manual of Forestry. The information of most value to American foresters is con- tained in miscellaneous books, pamphlets, bulletins, and forest periodicals, but the material is scattered and often not available when desired. The German and French text-books are excellent. The aver- age American forester, however, cannot read German and French 8 FOREST MENSURATION. with sufficient facility to use these works. Moreover, these books do not contain a great deal of matter which an American forester must know. Every forester engaged in research should learn to read German and French in order to follow the results of foreign work in forest Mensuration, much of which is of great value to us, but which cannot be entirely included in _text-books. The best foreign books are the following: Die Holzmesskunde, by Franz Baur, Lehrbuch der Holzmesskunde, by Udo Miller, Leitfaden der Holzmesskunde, by Adam Schwappach, Cubage et Estimation des Bois, by Alexis Frochot. Of these Schwappach’s book is the most useful to the average beginner. Surprise is often expressed that the British foresters in India have not developed a more extensive literature, and that what has been written has so little reference to the practice in India. The reason why there is so little Indian literature on forest mensuration is that there has been relatively little work of research carried on in the study of growth. There is in India no central department of research, and practically all that has been done in the study of growth has been in the preparation of working plans; and asa rule the data of growth are based on relatively few measurements. The methods of mensuration, working plans, and silviculture in India are of great value to American foresters in the Philippines. It is unfortunate that a more complete account of these methods has not been presented together in a series of books. A list of books on forest mensuration is given in the Appendix. 4. Units of Measurement.—The ordinary American units are used in the work of forest mensuration in the United States. The diameter measurements are taken in inches. Lengths of logs and trees are measured in feet. For ease in averaging and comparing the results of measurements, inches are divided into INTRODUCTION. 9 tenths instead of eighths, and feet are divided into tenths of feet instead of into inches. It is not necessary or desirable at the present time to use the metric system in forest work in the United States. Although clumsier than the metric system, the ordinary American units may be made fully to answer the requirements of the most exacting scientific work. ‘The metric system has many advantages over the present American system, but the results of measurements. t@ken in the woods, if expressed in the metric units, would be unintelligible to most persons for whose practical use the figures are designed. This does not apply to the Philippine Islands, where the metric system has already been introduced with success. The unit of volume most commonly used in America in the lumber industry is the board foot. In small transactions, standing timber is often sold by the lot or for a specified amount per acre. Standing trees which are to be used for lumber are occasionally sold by the piece. Hoop-poles and other small wood are sold by the hundred or thousand. Ties, poles, piles, and mine props are sold by the piece, the price varying acording to specifications as to diameter, length, and grade, or by linear feet. Fire-wood and wood cut into short bolts, such as small pulp- wood, excelsior wood, spoolwood, novelty wood, heading, etc., are ordinarily measured in cords. In the Adirondack Mountains the 19-inch standard, or as it is often called, “market”, is a common unit of log measure. In some localities a log 22 inches in diameter at the small end, and 12 feet long, is used as a standard log and is the unit for buying and selling timber. In other sections standards are used which are based on logs of other dimensions, as explained in a later chapter. 3 In New England wagon stock is sometimes sold by the cubic foot, but the unit is not commonly used in commercial transac- tions in this country. When used, it is employed to measure long timber, and the results are calculated for the square sticks rather than for the full contents of the round logs. The cubic 10 FOREST MENSURATION. foot, however, is commonly used in measuring precious woods imported from the tropics. Such timbers are also sold by the ton. Formerly the Spanish cubic foot was used in the Philip- pines, but the cubic meter is now the standard unit, as estab- lished by the Forest Act of 1904. CHAPTER .t. . THE DETERMINATION OF THE CONTENTS OF LOGS IN BOARD MEASURE. 5. Definition of Board Measure.—The unit of board measure is the board foot, which is a board one inch thick and one foot square. The contents of a board one inch thick are equal to the number of square feet of surface of the board’s side; hence the term superficial contents which is applied to the number of board feet in lumber. Board measure is used also for measuring the contents of lumber of other thicknesses than one inch and other widths than one foot, as plank and scantling. The ex- pression superficial contents, which originally was applied to inch boards, is now popularly applied to the number of board feet it’ lumber of any dimensions whatever, and also to the contents, board measure, of round logs. The number of board feet in any piece of lumber is obtained by multiplying the product of the width and thickness in inches by the length in feet and dividing by 12. It is necessary to divide by 12 because the width of the board is expressed in inches instead of feet. Thus a 2X4 scant- 2XAXK IZ 12 are constructed to show the contents of boards and scantling of all commercial dimensions. Such tables are published in a variety of forms, usually in so-called Ready Reckoners, such as those mentioned on page 369. In some localities board meas- ure is based on boards 7 of an inch in thickness. In this case the unit is a board one foot square and 2 of an inch thick. Con- TE ling, 12 feet long, contains ( )=s board feet. Tables 12 FOREST MENSURATION. siderable confusion has resulted from this deviation from the usual unit. 6. Board Measure Applied to Round Logs.—In America the contents of round logs are usually measured in board feet. This measure does not show the entire contents of logs, but the amount of lumber which it is estimated may be manufactured from them. ‘The contents of any given log are determined from a log table showing the estimated number of board feet which can be cut from logs of different diameters and lengths. Such a table is called also a log scale or log rule. The method of constructing a log table or rule is to reduce the dimensions of perfect logs of different sizes, to allow for waste in manufacture, and then to calculate the number of inch boards which remain. 7. Principles of Constructing Log Rules.— There are, in general, five methods of constructing log rules. 1. The method of diagrams. Full-sized circles of all diam- eters are drawn on large sheets of paper. Lines are drawn across to represent the boards, each being separated by a narrow band representing the saw-kerf. As many boards are fitted into the diagram as possible, assuming a specified minimum width of board and a reasonable width of slab. The contents in board feet are then calculated from these diagrams for logs of all lengths and diameters. 2. Mathematical formule. In this method a mathematical formula is used which reduces the dimensions of the log by an amount proportionate to its size, to cover the loss in slabbing, edging, and sawdust, and gives directly the amount of lumber in board feet. As described later these formule are, with a few exceptions, mathematically incorrect, and tables derived from them are of little value except as a rough approximation which is little better than the ocular estimate of a trained sawyer. 3. By the results of actual experience at sawmills. A number of log rules have been constructed from the results of actual tests at the mill. Logs of different sizes are followed through the mill CONTENTS OF LOGS IN BOARD MEASURE. 13 ‘and the product of each ascertained. The results of a large number of such measurements are averaged together in the form of a log table. 4. By correcting existing log rules. Lumbermen frequently change some log rule which has proved unsatisfactory, by apply- ing corrections to make the results conform with what their own mills actually yield. 5. By first constructing a rule in«standards (see page 53) and then translating to board measure by applying a uniform converting factor to each of the values in the standard rule. As shown later, this method is incorrect. 8. Conditions Influencing the Contents of Logs in Board Feet.—The amount of lumber which, actually can be cut from logs of a given size is not uniform because the factors which determine the amount of waste vary under different circumstances. These factors are as follows: , 1. The thickness of the saw. A thick saw causes a greater waste in sawdust than a thin one, as, for example, a modern band- saw. The old-fashioned rotary saw cut a kerf } or % inch thick, while the modern band-saw cuts out } inch. 2. The width of the smallest board which may be used. The narrower the board that can be utilized, the greater will be the total product of a specified log in board measure. If the width of the narrowest board that can be used is 6 inches, there is greater waste in slabbing than if 3-inch boards can be taken. In the former case a thick slab is thrown away which at its center is thick enough, but which is not wide enough, to make a board _ 6 inches wide. It is obvious that such slabs often would saw out a 3-inch board. 3. The thickness of the boards. A log sawed into 2-inch plank will yield a greater number of board feet than if sawed into inch boards, because there are fewer cuts and hence less waste in sawdust. If the boards are cut less than one inch in thickness, there would be even a greater sawdust waste than by sawing 1-inch boards. 1} FOREST MENSURATION. 4. Skill of the sawyer. Great judgment is required to cut timber in such a way as not to waste any material besides the absolutely necessary slab and saw-kerf. As a rule the sawyer must decide very quickly how a given log should be sawed, so that he must be a man of clear head, quick perceptions, and great experience. Mill-owners find it economy to hire the best available sawyers, even at very high wages. 5. The efficiency of the machinery. It is obvious that a saw which is poorly set and filed will waste more material in saw- dust than one properly handled. Cheap machinery often pro- duces boards thicker at one end than at the other. This means that the total product from a given lot of logs will be less than could be obtained from better machinery. - 6. Defects in the timber. Very few logs are perfect. In the majority of logs there is more or less waste due to crooks, rot, knots, worm-holes, or other defects. | 7. Amount of taper. It is evident that logs having consider- able taper will yield more than those with little or no taper, since some short boards may be cut from the wider portion of the logs. 8. Shrinkage. Boards shrink to a certain extent after sawing. 9. Value of Board Measure Applied to Round Logs.—The lack of uniformity in. the conditions influencing the contents of logs in board feet has led to wide differences of opinion as to how log rules should be constructed. It is obvious that a rule based on a 1}-inch saw-kerf and a given allowance for waste by edging does not give accurate measure when used with a saw of different gauge and in a mill which wastes less wood» in edging than allowed by the rule. Many lumbermen have not been satisfied with such log rules as have been constructed, and have devised new rules to meet their special local requirements. The multiplication of rules has continued until there are now over forty in use in this country and Canada. The large number of rules, their inconsistencies, and the incorrect methods used CONTENTS OF LOGS IN BOARD MEASURE, 15 in applying some of them have led to great confusion and incon- venience; and in some cases landowners have been defrauded through the use of defective rules. One of the sources of difficulty in log measurement is the lack of uniformity in applying any given rule. Most rules require the measurement of diameter at the small end of a given log, an slabbing. Inasmuch as the taper is thus left out of consider- ation, a single long log will show a different product, wit: a given log rule, according to the lengths and arrangement of the short logs into which it may be cut. In some cases, of course, a long log may be divided only in one way because of crooks or other imperfections which have to be avoided. But with a perfectly straight long log there may be a dozen or more methods of arranging the short lengths, and in almost every case a different product will be obtained from the log rule. This is well illustrated by the example of two pitch pine trees measured in Pennsylvania, each having a merchantable length of forty-two feet (see table, page 16). This table shows a variation of the scale of tree No. 1 between 149 and 181 board feet, and of tree No. 2 between 136 and 167 board feet, according to the method of cutting. The largest scale is obtained by cutting four logs. In practice, however, it would frequently be more profitable to cut three logs with a smaller scale, because of the greater expense in handling the larger number of logs. Appreciating the facts explained above, one is inclined to advocate the abolishment of the board foot as a unit in measuring round logs. The board foot is, however, an exceedingly convenient unit. It is much simpler for the millman to know at once the actual product of logs than to be obliged to calculate the product by applying converting factors to some other unit like the cubic foot or the standard. In estimating the yield of a given tract in saw timber, in purchasing stumpage to supply a mill for a given length of time, in valuation of forest land and similar work, on the assumption that the increase at the other end is lost 16 FOREST MENSURATION. Tree lI. Tree II. Tree I. Tree II. Length} Total Log Length Total ic Length} Total Length} Tota] Log | of Toc! Scz ; - . Tone ee Log 7 : No: | race®"| Bact. | No | Seecet” [Bartel] Ne | Srccte”| Bare. | No | Geet” saci’ I 16 I 16 I 16 I 16 2 14 2 14 2 16 2 16 3 12 153 3 12 153 3 10 157 3 10 157 I 14 I 14 I 10 I 10 2 16 2 16 2 16 2 16 ? 3 12 149 | 3 12 | 149 || 3 16 157 ‘i oe 16 | 143 I 12 I 12 1 16 I 16 3 14 2 14 2 10 2 10 3 16 162 3 16 136 3 16 153 3 16 144 I 10 I 10 I 14 I 14 2 10 2 10 2 14 2 14 3 10 3 10 3 14 157 3 14 144 4 12 17! 4 12 161 | —— — —— -- —_| ——— | ——_—_|—__-| - |__|] 1 16 I 16 I 10 I 10 2 12 2 12 2 10 2 10 3 14 159 | 3 14 | 148 3 12 3 12 SO OOO OOOO 4 10 175 4 10 165 I 14 I 14 ——| ———] — —|— -}| 2 12 2 12 I 10 I 10 3 16 I5I 3 16 140 2 12 2 12 ——_| ——__ |__| —__|-_—- 3 10 3 10 I 12 I 12 4 10 179 4 10 167 2 16 2 16 — —| — ——|——|— ———=|| 3 14 167 3 14 ro I 12 I 12 2 10 2 10 x 10 3 10 4 10 181 4 10 158 | — ——— board measure has proved a very useful unit of contents and will not be given up. | While board measure is practical for measuring logs used for lumber, it is an unsatisfactory and incorrect unit for comput- ‘ing the contents of logs in shingles, lath, staves, shakes, and so on. At present lumbermen estimate roughly the number of shingles, lath, and staves contained in a thousand feet of lumber. It would be a great advantage if there were log tables showing the product in shingles, staves, etc., of logs of different diameters and lengths, just as board-foot tables show the contents of logs in inch boards; \ CONTENTS OF LOGS IN BOARD MEASURE. 17 but as far as the author is informed, no attempt has ever been made to construct such tables. Board measure is extremely unsatisfactory for measuring the product of trees or logs in pulp-wood. Practically the whole log is used in making pulp. A solid measure should be adopted, like the standard or the cubic foot for the measurement of pulp- wood, dye-wood, etc., where the whole log is utilized. As the value of wood increases the use of the cubic foot will undoubtedly be increased and eventually to a large extent replace the board foot as unit for logs. 10. Need of Uniformity in the Measurement of Logs.—The large number of log rules and their inconsistencies have led to a wide-spread demand for a reform which will lead to uniformity in the methods of measuring logs. There is a demand for a standard log rule which may be used in all cases of dispute and which will replace the present rules in purchasing and selling land and timber. A universal log rule is needed not only in commercial transactions, but also in scientific work of forestry, notably in investigations of the volume, growth, and yield of trees. The Government has in recent years published volume tables of standing trees, tables of growth in volume of different species, and tables showing the yield per acre of timber in different regions. Some tables are based on the Doyle, some on the Scribner, some on the Maine rules, etc. A scientific comparison of these tables is impossible because differences in results may be due to the variable differences in log rules used in computing the contents of the trees, rather than to differences in laws of growth. It will be impossible to construct a single log rule for com- mercial use which would be satisfactory to all lumbermen, with- out modifications to meet local conditions of the forest and of manufacture. Such a rule is not proposed by the author. It is, however, perfectly possible and practical to construct a log rule showing the product of logs under the best conditions, to serve: 1. As the recognized commercial rule under the best con- ditions of lumbering. -— 18 FOREST MENSURATION. bo . As a standard in cases of dispute. 3. As a standard of comparison in all scientific work. 4. As a foundation for local commercial rules. Local rules can never be altogether dispensed with as long as the board foot is used as a unit in measuring logs. A local rule showing the average yield of logs of a’ certain species from a certain region under given conditions of manufacture is very useful timber in estimating, and it is probable that many lumber- men will insist on local rules for buying and selling logs. Thus it would be necessary to have a local log rule for the hardwoods of the southern Appalachians, another for the second-growth timber of New England, and so on. Local log rules are necessary just as local volume tables are necessary, as an aid in estimating and, if desired by the parties concerned, for use in timber sales; but there should at the same time be a standard which is just as definite as would be a recog- nized method of measuring logs in cubic feet or cords. 11. Principles of Constructing a Standard Log Rule.—Numer- ous attempts have been made to standardize a log rule, as is shown by the, State legislation on the subject. ‘These attempts have tended to increase the confusion because the States have adopted different rules. So far the efforts have been to select a rule from those in common commercial use. Each of these rules, however, has serious defects which makes it unfit to be a standard, and it is these defects which have prevented lumber-men from reaching an agreement as to which is the best rule. The question will not be settled until lumbermen agree as to the purpose and the requisites of a standard rule for measuring logs. The requisites of a satisfactory standard rule are, in the 4 author’s opinion, the following: t. It should show the product of logs in boards one inch thick. ~ 2. It should be based on the use of a saw cutting a specified kerf, for example } inch. CONTENTS OF LOGS IN BOARD MEASURE. 19 3. It should include boards down to a specified minimum width, for example 3 inches. 4. Its construction should presuppose the use of good machinery and skilled sawyers. 5. It should be based on logs normally straight and sound. Logs are seldom absolutely perfect, there being nearly always some loss by crook or hidden defect. Logs normally straight and sound are then the best which are commonly found, in con- trast to ideal logs which are scarcely ever encountered. An allowance in the rule should, therefore, be made to cover this loss which is practically always present. 6. It should be based on correct mathematics. This point is mentioned because many rules are mathematically unsound, as, explained in the discussion of the various rules. 7. It should be based on a formula rather than on diagrams or mill scales. The objections to the use of diagrams are: (1) the values in the table increase from diameter to diameter by irregu- lar differences; (2) the values cannot be easily corrected to conform to new or special conditions of manufacture; (3) the values cannot be easily checked as to correctness. The method of constructing log rules by tallies from a mill scale is undesirable, because it would be almost impossible to conform to the con- ditions and requisites mentioned above. Such a rule would almost inevitably bear the impression of local quality of timber and local conditions of manufacture. It would have temporary and only local value; and when it should become necessary to alter it to meet new conditions, the work would have to be done all over again. The rule should be based on a mathematical formula by which the volume of the whole log, the loss by saw-kerf, slabbing and edging, and the loss by normal defects, are separately deter- mined. The amount allowed in the formula for loss in manu- facture should not be determined theoretically, but by tests at the mill. Such a formula can be corrected for different widths of saw, special dimensions of the product, and so on. At any 20 FOREST MENSURATION.,. time one can check the values in the table or himself construct a table. But the most satisfactory feature of the method is that the exact allowance for waste in slabbing, saw-kerf, crook, etc., is known, and there is an opportunity to discuss and agree on these points, so vital to the success of a universal rule. 12. Selection of a Standard Rule.—None of the rules now in use met the requirements of a satisfactory standard rule. Two independent investigations, however, have recently been made which have clearly demonstrated the correct method of con- structing log rules, and which should lead to the desired harmony in log measurement. The studies of Prof. A. L. Daniels of Bur- lington, Vt., and Prof. J. F. Clark of Toronto, Ont., constitute the most valuable contributions to the subject of log measure- ments that have been made. Each has proposed a formula to construct a standard log rule, based on the same general mathe- matical principles and designed to accomplish the objects of a | standard rule described in the preceding section. The formule differ mainly in that Prof. Daniels makes no allowance for loss by crooks and disregards the taper of logs, and Prof. Clark makes a greater allowance for defects, including normal crook, and provides for a taper of one-half inch in every four feet of length. The differences between the two rules are not great, but in the author’s opinion neither can be definitely proposed as the universal standard without some further study. Thus Prof. Clark’s table appears to meet the requirements of the standard rule so far as pine is concerned; but it remains to be proved whether his allowance for crook and taper are applic- | able to southern hardwoods, cypress and other trees than pine. Prof. Daniels allows less for normal defect than Prof. Clark and this point in his formula would have to be proved by mill tests which have not yet been made in sufficient numbers. The points of difference between the two rules are capable of proof by tests. The establishment of one of these rules or of some other rule as a universal standard can be brought about only by agreement among the lumbermen and foresters, and by sub- CONTENTS OF LOGS IN BOARD MEASURE. 21 sequent enactment of the necessary legislation giving legal sanc- tion to the rule. 13. Construction of Local Log Rules.—It was explained in section 1o that local log tables will be required in estimating and valuing timber and for many commercial transactions. At the present time the old rules are used for this purpose. The mill-scale studies which are being made by the U. S. Forest Ser- vice are, however, proving the old rules to be inaccurate, and in many cases unsuited even for local use. It is probable that nearly all the old rules will have to be superseded by new local rules. Where the necessity for a new local rule is proved, the new one should not be made directly from the average of the mill-scale tallies, but should rather be based on the universal rule by applying to it corrections which are determined in the local miil studies. Local log tables are necessarily more or less temporary in character, differing from the standard rule in the allowance for defects, crook, saw-kerf, thickness of slab, and so on,—factors which will in time be changed. Local log rules will be constructed by adding to or subtracting from the values in the standard rule a certain percent or per- cents, the latter being determined by local mill studies. 14. Graded Log Rules.—Graded log rules show the product of logs in lumber of different grades and value. A number of such tables have been made up by the U. S. Forest Service in con- nection with the preparation of graded volume tables. None have as yet been published. They promise to serve a useful purpose in the valuation of logs. 15. Log Rules Sanctioned by Statute.— There are in the United States six different log rules sanctioned by State law, as follows: The Doyle rule, which is the statute log rule of Louisiana, Florida, and Arkansas; the Scribner rule, adopted by Minne- sota, Idaho, Wisconsin, and West Virginia; the Spaulding rule, adopted by California; the Vermont rule; the New Hampshire rule; and the Drew rule, adopted by Washington. To this list may be added the New Brunswick, the Quebec, and the British bo Le) FOREST MENSURATION,. Columbia rules of Canada. ‘The U.S. Forest Service has adopted the Scribner rule (Decimal C). Several lumber associations give their official sanction to certain rules; for example, the Doyle-Scribner combination rule, adopted by the National Hardwood Lumber Association, and the Drew rule, sanctioned by the Puget Sound Timbermen’s Association. This official approval of different log rules only adds to the general con- fusion. 16. Which is the Best Log Rule in Common Use.—From the standpoint of accuracy, the Maine rule, if used with short logs, is probably the best of the Eastern rules in common use. The present methods of using it to measure long logs give unsat- isfactory results, but this is due to the incorrect application of the rule, and not to defects in the log table. The Spaulding rule appears to give satisfactory results for the Pacific Coast con- ditions. 17. Early History of Log Rules.—The board foot as a unit of measure for sawed lumber has been used in this country for a great many years. ‘Thus the measurement of the superficial contents of boards is described in A Complete Treatise on the Mensuration of Timber, by James Thompson, published in Troy, New York, in 1805. At that time, as shown in this same work, round logs were measured entirely in cubic feet, by the old Fifth Girth Formula, brought over from England. In the book above mentioned there is no reference to dog tables or to estimating the contents of logs in board measure. ‘The earliest mention of a log rule for board measure, known to the author, is contained in A Table for Measuring Logs, Anon., Portsmouth, Me., 1825. The rule, as described in this brochure, is as follows: “Cast + of the diameter of the log and then reckon as many boards as there are inches diameter, and the width of boards the same. For example, take a log 12 feet long and 12 inches through at the top, and by casting off + as above mentioned it leaves 9 inches, which I call 9 boards g inches wide and each board makes 9 feet; and then multiply the number of boards by the number CONTENTS OF LOGS IN BOARD MEASURE. 23 of feet in one board and the product is 81 feet; and by the same rule you may cast any log whatsoever.” It will be seen that this is the same as the square of the three- fourths formula described on page 49. One of the oldest formulae for determining the board con- tents of logs is shown in The Mechanic’s Assistant, by D. M. Knapen, published in New York in 1849. The rule is as follows: “If the log be 2 feet in diameter, or less than 2 feet, allow 2 inches on four sides for the thickness of the slabs, and one-fifth for saw-calf, and one board for wane; but if the log is more than 2 feet in diameter, allow 3 inches for the thickness of each of the four slabs, and one-fifth for saw-calf, and two boards for wane. If, however, the logs are very straight and smooth, the slabs may be thinner. ”’ For so-called market boards the rule reads: “Market boards are usually a little less than one inch in thick- ness; and consequently the number of feet of market boards in a log will be greater than the number of feet of inch boards. To find the number of feet of market boards, in any log, allow one- eighth for saw-calf, and apply the above rule for inch boards with this difference.” About this same time there appeared a rule giving the same results as the present Doyle rule, reading as follows: “From the diameter of the log, in inches, subtract 4 for the slabs. Then multiply the remainder by half itself, and the product by the length of the log, in feet, and divide the result by 8: the quotient will be the number of square feet.” © This rule was published for the first time, as far as the author is informed, in Elements of Drawing and Mensuration, by Charles Davies, New York, 1846. These facts are interesting in view of the introduction of the Doyle rule between 1870 and 1880, which gives exactly the same results as this old rule and which was claimed to be new. Undoubtedly Mr. Doyle’s formula was original, but an equivalent table had been used thirty to forty years before. : “a Six TEEN- Diameter Name of Rule. 6 | 8 | 10 | 12 | 14 | 16 | 18 Board CN UEEND, 5 aia s 0. va 9x 43 70 105 146 193 247 P= Piroereih. -f5. en. avd ess 2 57 89 127 172 223 MPI oe ciate Sais: okie td 32 54s) 79 114 159% | 213 0) a eee 16. 36. 64 100 144° | 196 ~ Holland or Maine....... 44 68 105 142 179 232 PION as ghk vied mr wis eee 32 51 80 117 160 213 PSPATHCLITNG, acs > Siscpn > no «| = 0 cea ee 50 77 114 161 216 ~ New Hampshire........ 35 54 78 106 139 176 7 - Humphrey or Vermont. 43 66 96 130 | 170 | 257 UIA OOT. yo scah os akin ee a 41 69 100 137 182 238 Cumberland River. ........]. .... -/s}s «een 47 68 93 121 153 PROG. ectin i 9 nisin Shin,0 5 + xn be aml 5 ee 64 98 142 | 197 Faget mes teas Moai | sialic a's 34 56 84 117 156 200 Square of three-fourths. . 48 75 | 108 147 | 192 | 24g. © Square of two-thirds. ...|......|...-.- 58 85 114 150 192 seit, \-3, ee Oe ere No vajlues gijven PURI sis x -cia 5 «8 abs Eis ote 25 49 77 107 | 142 | 383 ISN dela cigs 8 32 59 80 120 160 eer COMIN DID. |. as 6-> te asnteg ph aeals pas 55 84 119 160 Mlnae PTUSWICK, .. ss os. Ts acicdalecectinlenal 96 130 |40ez. PIBSCH DCE. 6 44's iv s'om cle penne eee 42 68 100 136 Grange Baver. 5.5... sabe ss - 00} s ee ae 76 104 136 NE ic, sso ee Ss CRESS CoE Eee 64 ) 84 112 144 WOPtN WESECER: 3. cos: vecucdie es wate 33 61 77 117 170 PICT: ».1's weiss aie oes 28 49 75 110 148 195 Partridge. . seiccubices ia. 26 46 68 102 140 | 180 ae «= sinister ea 21 41 64 100 140 179 2) 5 SAP ares Paces Pes 69 109 157 PC TUPWUIL: > .. Sih Begg wit aie ares Sha aeons 65 Baughman’s rotary saw .|_ 17 41 70 Baughman’s band-saw...} 20 41 73 r SSH RIVER ES. iain ty. <0 26 49 75 OU cohapiid site 22 40 61 WSO sis'idde coh wal 23 46 67 WEUCORS os ses's re Ce ere ee Warner. '..'5\.. Wistdels'’s gwapumae® 30 40 SOROTILORD, 5.5 nn'n'yie. pss insw vl oc 32 60 EE hrs choi wha. o 5's No values} bivest |? BEEF TCG c ito Gn-« wincgin’s <2 3, Nong 125 | 168 | po ESS A se enn “ae tf Tras) 1G Finch and Ack API EAs ores Cee 112 157 COncta rene ios sew caters oi arg | 2688 VA ORR APE CREE SR a 128 167 Wheelews 5 S.Chue eats 132 174° ‘International (band-saw) 150 200 — FOREST MENSURATION. COMPARISON OF LOG RULES * Values for 6, 8, and ro inches are those use . t Values read off from a scaier's stick. f CONTENTS OF LOGS IN BOARD MEASURE, 25 s FOR BOARD MEASURE, ed FOOT LoGs. ¥ in Inches. 20 | 22 | 24 | 26 | 28 | 30 | 32 | 34 | 36 | 38 40 Feet. PA aR a 308 | 376 | 450\| 532 620 714 814 923 | 1038 | 1159 | 1287 282 | 347 | 419 | 497 582 674 773 878 990 | 1108 | 1234 280 | 334 | 404 500 582 657 736 800 923 1068 | 1204 256 | 324 | 400 | 454 | 576° | 676.| 784 | goo | 1024 | 1156 | 1296 302'| 363 | 439 | 507 | 614 706 | 795 | goo | 1026 | 1135 | 1261 O92. -236 | 416.) 501 576 656 741 832 933 | 1066! 1200 276 | 341 | 412 | 488 569 656 748 845 950 | 1064 | 1185 217 |. 262 | 313 367 426 489 557 628 704 785 870 267 | 320 | 384 300 | 369 | 444 | 526 | 609 | 697 | 792 |: 892 199 | 229 | 268 | 320 372 427 485 548 614 685 759 248 | 324 | 392 | 476 | 562 | 632 | 725 | 845 | 920 | 1037 | 1160 250 | 305 | 366 | 432 | 504 | 582 | 665 | 754 | 848 300 | 365 | 432 | 507 | 588. | 675 | 768 | 867 | 972 236 | 285 | 341 | 400 | 464 | 533 | 605 | 684 | 768 | 854] 946 below} 20 fejet. 230 | 284 | 344] 411 | 485 | 567 | 655 | 752 | 857 | 963] 1067 286 | 347 | 420 | 507 580 673 760 867 947 1040 | 1173 261 | 320 | 386] 457 BiG 619 708 804 906 | 1015 | 1129 300 | 362 | 432) Y/ 229-| 285 | 346 414 | 487 | 567 | 652 | 744 | 841 945 | 1054 213941).258. |, 308 |. 360 418 480 546 616 692 769 853 245°) 204 | 374-1 465 563 666 TTL 896 | 1027 | 1161 | 1296 248 | 324 | 392 450 536 632 pe 845 920 1037 | 1160 See soo! 430 | 512 | 593 |680..] 793-| 872 4 977 F 288 | 350 | 416 | 486 564 650°] 738 834 998 300 © 366 | 433 506 600 705 ate sao.) AL3 | 493] 579 | 6724 Sa7n 877. | W989. | 107 |sr282 261 | 320} 385 | 456 | 533 | 588 | 675 | 768 310 | 382 | 457 | 540 | 633 | 722 | 822 | 934 | 1054 | 1142-| 1294 340 | 417 | 500 | 590 686 790 goo | 1022 | 1182 | 1286 | 1425 302 | 366 | 436 | 513 | 590 | 674 | 771 a, 280 306 | 374] 448 | 529 616 733 814 922 Bee pte taai 3731) 446 | 513 | 502 | O73 |. 754.4 853 973 | 1120 203] 2ne| 316 | 372 431 490 560 630 366 | 322 | 384 |] 450 522 logs} over/15 felet —lo|ng. 275 | 341 | 415 | 498 590 691 803 925 | 1058 290 | 338 | 402 | 492 | 575 | 649 | 728 | 797 258 | 318 | 400 | 474 552 624 733 840 | 928 | 1054 |] 1181 416 | 507 | 603 | 708 | 821 | 942 | 1072 | 1210 | 1356 | 1511 | 167% 261 | 316 | 377 | 4ar 512 588 | 669 277 | 337 | 404 | 475 | 553 | 636 | 725 320 | 390 |*47 555 | 645 | 745 | 850 | 965 | 1085 | 1210 | 1345 ~ SS SE EEE ee eee ee eee eee ee Santa Clara Lumber Company, New York. }Y av 26 FOREST MENSURATION. The exact date when J. M. Scribner first published his log table is not known to the author. ‘The fourth edition was issued in 1846. It is probable that the rule is one of the oldest used in the country. It is impossible to determine exactly when the board foot camey into general use as a unit for measuring logs. Probably it was— not generally used before 1820, for the works on mensuration printed before that date make no mention of log tables. It is probable that about the middle of the century a number of different rules were constructed in different parts of the East, including the Younglove, the Parsons, the Saco River, and other rules. The conception of using a standard log as a unit of measure is also very old. The 24-inch standard is described in Davies’ Elements of Drawing and Mensuration, mentioned above, and . reference is made to the 1g-inch, 22-inch, and 24-inch standards in Knapen’s Mechanic’s Assistant. This point is also inter- esting as showing the custom in the early days of cutting logs 13 feet long. 18. Comparison of Log Rules.—The different log rules are com- ‘pared in the table on pages 24, 25, which shows the board contents of sixteen-foot logs of different diameters. This table, including as it does all the log rules, is presented for the convenience of those wishing to make:comparisons between the values obtained by different rules. It should be borne in mind, however, that some of them are really not comparable. For example, the Constantine rule can hardly be compared to the Cumberland River rule, because the former gives the whole contents without allowance even for a sawdust waste, whereas the latter is intended to cover a.special amount of defects such as occur in river _logs. CHAP iS I11. DETAILED DISCUSSION OF LOG RULES. 19. The Champlain Rule.—This rule was devised by Prof. A. L. Daniels of the University of Vermont. The following de- scription is based on certain portions of Bulletin 102 of the Vermont Agricultural Experiment Station, entitled “The Meas- urement of Saw Logs,’’ and on private correspondence with Pro- fessor Daniels. In many places the author has used Professor Daniels’ own language. The Champlain log rule is developed in the following way: It is assumed that all logs are straight, round, and free from defects, and that the loss in the manufacture of the board is due only to sawdust, slabbing, and edging, and not to crooks, knots, or other blemishes. The only portion of the log dealt with is that which will make boards. ‘The thickness of the slab is based entirely on the diameter at the smallest end, the taper of the log being disregarded; in other words, the log is considered a cylinder whose diameter is the same as the average diameter of the top end of the log. We begin, therefore, with a cylindrical log, round, + straight, and of perfect quality. The solid contents in cubic feet of such a log is determined by multiplying the area of the cross-section : D? in square feet by the length, that is, by the formula es XL or V=0.7854 xX D2XL, in which V is the cubic volume, D the diameter, and L the length of the log. If Dis expressed in inches and L in feet, it is necessary to divide the result by 144. If the log in question is 12 feet long, the formula reads V =0.7854 X D? +12. 27 28 FOREST MENSURATION. This result may be translated into board measure by multi- plying by 12 on the assumption that each cubic foot contains 12 board feet, which is the case if the waste in sawdust, slabs, and edging be disregarded. ‘The solid contents of the log in board feet is, therefore, obtained by the formula Bj = (0.7854.D?L +144) X12, or for a twelve-foot log Bf =0.7854D?. It will be seen by reference to section 29 that this is the same as the Constantine rule. In the manufacture of boards two allowances for waste must be made, one for saw-kerf, and the other for slabs and edging which may be called surface waste. Consider first the allow- ance for saw-kerf. Suppose that the log is sawed through and through by the method sometimes called ‘“slash-sawing,”’ or “saw- ing through alive”; suppose, further, that a circular saw is used which cuts out a kerf one-quarter of an inch wide. The loss in sawdust for the log will then be one-fifth of the contents. That is, for every inch board there is a loss of one-quarter of an inch. There remains, then, four-fifths of the solid contents. Since the con- tents of a 12-foot log is 0.7854 D?, there remains after sawing 4X0.7854D? or 0.62832D?. This represents the exact value of the untrimmed inch boards, including slabs, in a perfect log. It will be seen, however, that if the saw had cut a wider or narrower kerf, the numerical factor would have been different. In case of a band-saw which cuts a kerf of } inch, the contents would be : x SEAIOP or 0.6981 3D2* The waste by sawdust has now been accounted for, but no * Expressed mathematically, a saw of any thickness, S, wastes S parts in Tao I SAG every 1+5S parts, and the contents of the log after sawing is Pee 6x 7 D?. DETAILED DISCUSSION OF LOG RULES. 29 allowance has been made for loss in slabbing, edging, and for such slight imperfections as are present in the most perfect logs. It is assumed that logs, whatever the diameters, have an average amount of waste in slabbing, edging, etc., proportional to the amount of surface. The amount of surface is proportional to the diameter and length. That is, a 24-inch log of a specified length has twice as much surface as a 12-inch log, and therefore the waste in slabs, edging, etc., in the former is just twice as great as in the latter. In the same way, a 16-foot log of a specified diam- eter has twice as much surface, and therefore twice as much waste in slabs, etc., as an 8-foot log. In order to construct a log table, then, it is necessary to find the relation existing between the surface waste and the diameter. This proportion is obtained from the evidence of sawyers and scalers and by using diagrams as a check. After extensive investi- gation, Professor Daniels has concluded that the surface waste in perfect logs is equivalent to an inch board having a width equivalent to the diameter of the log; that is, for surface waste he subtracts as many board feet as there are inches in the top diameter. The complete formula, on the basis of a saw-kerf of + inch, is Bj =(0.62832D?2 XL +144) X12—D, This formula may be used for any length and diameter, or the contents of 12-foot logs may be first determined and the values for other lengths obtained by multiplying by the length and dividing by 12. The formula for 12-foot logs is Bj =0.62832D? — D. The intimate connection should be noted that exists between this rule and the Constantine rule. Take four-fifths of the Con- stantine values, subtract the diameter, and one has the Cham- plain rule. Or add to the figures in this rule the diameter, in- crease by one-fourth, and one has the Constantine figures, which of gtg Og 6sL F1Z 699 zg og ces 16¢ gtr cor LS zit Or 6c 162 6tZ Lol 999 tz9 gs It 66+ gor git cle eee 162 6z gz gtl £69 gfg oz9g 1g zts £os Cor gzr Lge gre orf ELE Qe Lz zgg Ltg 119 cLS 6S os Lor IfV C6e 6ce uee Lgz 1Sz Lz gz 1¢9 g6s CoS ces g6tr Cor ceVy 66£ g9t zee 66z 99z fee gz Sc zgs igs 1c o6b og 6z+ gé6e ggt Lee got 9glz Chz Fiz Cz $z ges Los glt oSt cer F6e 99t gee o1f ZQz Sz Cz L61 vz tz o6F tor gtr civ Lge 19¢ cee 60£ tgz gSz zz goz OgI x4 zz Ltr Lz oor gle ese 6z¢ got ZQT 6Sz C¢ez zie SgI CcgI oe Iz Sot ge ge Ive oze 66z kkz gSz Ccez C1z z61 ILt 6r1 Iz 3 oz 99t Lee gz got 6gz oLz 482 1fz ZZ £61 Li Sr CLI oz - 61 6ze zie +6z LlLz ogz tbe Czz goz o61 CL1 oS 1 6£1 IZI 61 xt gI b6z gl £gz Lez zz Liz 102 Q8I oll CCI 61 bzI gol gI x Li 192 Lez Lz 61z goz z61 gLI CgI ICI Ler re OI! 96 Li 3 QI 6zz Liz Coz £61 IQI 691 LS1 Cri a IZI 601 L6 tg gl = CI o00z o61 6L1 Sgr QS1 Lv1 Ler gzI QII Sor c6 tg tL C1 +I ELr FoI Cor oti of Ler SII -| 601 oor 16 ZQ SL to vat ~ C1 gtr obi ze ver YI 601 IOI £6 Sg gl ol z9 +S EI a zi bz1 QI II Sor g6 z6 Sg gl zl $9 6S zS oF is S II for 96 Z Lg Ig ol ol $9 09 +S 6h eV gt Il 2 oe RSS Py Oe TR, Ne A MOS BT oy eee | Mae OR he et Fests es, eres U8 or tg 6L Hs ol 99 z9Q LS e¢ gt tr ob ce If oI 6 99 €9 6S 96 z¢ 6+ St zh gf ce if gz bz 6 8 IS gh gt th ot ge ce ze of Lz $z 1Z 61 8 L gt gf $¢ ze of gz gz $z ZZ oz gl gI FI L 9 gz Sz tz zz Iz 61 gt Li CI +1 ZI II ol a s gI gl C1 +1 Cr zI II OI ol 6 g l 9 S ¥ 6 6 8 8 L L 9 9 S C + + ¢ + aes 6: 81 aq or et ia er ar IT or 6 8 ee ie - = *J99g ut 307 Jo yBue7q ae a) ‘ATNA DOT NIWIANVHS AHL ‘sjatueq “] ‘Vy jo uorsstiuied Aq pajutig ‘sjotuvd juowre’] pleqrory Aq ‘So6r ‘yy8tsAdod * | 09 gstt ror’ 61 rf gt 6z 2She 69Sz Sgt ZOTZZ Q10z CLQ1 ICQI sori b9Qz1 09 6$ ogre cO1t Cr1ok gtez 0992 tgbz Cotz QT1z oS61 CLL g6S1 611 1vzi 6S gs PSce got z16z Ivlz olSz 96¢z Looe gSoz Pegi Be zPs1 oLlf1 oozI gs LS Iie 9l6z 11gz gtgz ogtz Cree OS 1z P61 61g1 €Cog1 Seri tool QgSIr LS gf Ifof zlLgz SiLz ESSz £6Lz Pezz bLoz F161 GSZI COSt ott gLz1 LI 9S 6 gs 1z6z gglz S19e 1gtz Lotz Core 0007 CVI £6g1 6fC1 Vgc1 OfzI gLor cs = rs CIgz £99 61Sz IL coz vLoz gz61 gLLi ofgI 7TQVI Cer CgII L¢or +¢$ & gs o1lz ggSc Cztz €9Q7zz GLie L661 tCer Ae a | ey AS | Levi tgzi IVII 666 cs ze Logz oltz CLoTz g61z 6So0z 1z61 Peli Lror OISI ZLe Coz 9601 096 z$ S 1¢ CoSz ELce 1vzz OL1z LL61 org Cri €gci oStI QIcI Lert CSor £76 1¢ = of gotz IQzz boiz QzOz 1061 bLLi LYgl oz SI b6CI Lgz1 obii Fior L88 oS re) 6+ 11fZ 6g1z ggoz gor6r tzg1 Coli ISI ogti Qeer QIZI C601 £L6 1SgQ 6+ = Qt C1zz 660z C61 9981 oli €log1 gIS1 oor I Cgzi 99II oSor £f6 918 gt S LY Cz1z I10z oo61 ggLI 9lgI toSi eC Iver 6z7z1 LVET gool t6s zgl LY o got z~loz Cz61 QIgI I1ZI Fogt L6ov1 o6f1 Pgzi DLT oLol £96 9$g 6rZ ot Yn ee OO SS a (Sa eee cee | accuses | Nepetee eee e e ee ee® |— S) a e (ee Se . cr ao Itgi 6¢L1 gf gr ves Zev Off 1 Lzz1 Cz11 €zO1 o0z6 BIg gil cr a th LSQ1 6CLI 1991 toSt gori gorI olZ1 LUI CLOI LL6 Ogg zgl te9 bP 9 cv ILL glor CgSr zOti g6e1 Corr ZIZI 6111 QzOI z~¢6 6£8 gil Sg 7 S Z SggI oogI 11S1 zcvi ErCt vrz1 CSr1 9901 LL6 688 008 11Z ZZQ zv a Iv gogl CzG1 gt ri vSC1 69z1 CgII OOII SIOI 1¢6 greg 19L LL9 z6S if = ee SS eS Se ~ (ee ee | (ee ey a | ee | eee ee x oF QzSI gtri Loft Lgzi gozI Qz11 gbor $96 rekes Fog $zL ft9 oS ob i 6¢ Str CLEI g6z1 A grit 6901 £66 L16 ote tol Lg9 119 Cec: 6¢ Q gt glf1 Foe 1 IfzI 6S11 Lgo1 biol zb6 698 L6L vzl zSQ ogs LoS gt Le Col1 Cez1 99II L601 6zoI 096 z6g zg PSL 989 L19 6+S ogr Le gf ZETZI Lgorl Corl gfol £16 306 erg gLL Cr 6t9 teS 61¢ tor gt ce €or ZOII Ibol o0g6 g16 LEg 96L StL ¢Lo Z19 1o¢ o6+ “6zb ce ve 9601 gcol 196 €z6 £98 gog oSl z69 Clg LLS 61¢ zor tot t¢ gee ILOI LL6 €z6 898 Fig ogLl gol 1Sg L6S ets geht ber oge ¢¢e ze £96 916 S98 b1g ey Fe Ws z99 119 o9$ 60S Sr Lot 9S ce 1¢ Lo06 6Se 11g tol gif 899 oz9g ELS CzS LLt oft zee tee i£ 32 FOREST MENSURATION. give the number of board feet in the solid log. If the number of cubic feet are wanted, it is only necessary to divide the figures of the Constantine rule by twelve. Professor Daniels has devised two short rules of thumb which give nearly the same results as the Champlain rule. They are as follows: (A) Take five-eighths of the diameter, subtract one and multiply by the diameter. (B) Subtract one from the diameter, square, and the result is the contents of a log of that diameter 19 feet long. Both these short rules give slightly less than the Champlain rule. 20. Daniels’ Universal Log Rule.—In October, 1903, Professor Daniels published in Bulletin 102 of the Vermont Agricultural Experiment Station a new log rule which he called the Universal rule. The principles of constructing the Universal rule are exactly the same as for the Champlain rule, except that a greater allowance is made for waste in slabbing, edging, and for slight defects. ‘The endeavor was to secure a rule which would give the contents of logs of average grade. The Champlain rule is made for perfect logs while the Universal rule is applicable to second-grade logs which have slight crooks or other blemishes, such as are not of sufficient importance to be made subject to a special discount by the scaler. Professor Daniels gives the name “roughage” to the material wasted by slabbing, edging, and slight defects. After careful study he has given as an allow- ance for surface waste, or roughage, an amount equivalent to a 2-inch plank, whose width is the same as the diameter of the log in question. It will be remembered that this “allowance plank,” as Professor Daniels terms it, was, in the case of the Champlain rule, r inch in thickness. The Universal rule, expressed by formula, reads, therefore, Bf=(0.62832D? XL+144) X12—2D = (0.62832D? X L) +12 —2D, ai. ‘yt DETAILED DISCUSSION OF LOG RULES, 33 / or for 12-foot logs, Bf=0.62832D?—2D. Professor Daniels’ argument for choosing the value 2D as the width of the allowance plank is, in his own language, as follows: “The particular value 2, which I have chosen as the thick- ness of the allowance plank, adapts the rule to what might be called a second-class log. So far as I can find out it makes it just safe for a buyer to take a fair ordinary lot of logs without any dickering over premium or discount. If a lot of perfect logs were offered, it might be fair for the buyer to give a small premium. Just where premiums or discounts are called for must naturally be left to the experience of the men in the business. The main point and prime advantage in using this rule is that they need pay no attention to whether the logs are large or small since the rule is a level one. The choice, therefore, of this second factor (2D), whether it should be 1.5 or 2 or 2.25, is, after all, as much a matter of convenience as anything in the actual business of buying and selling lumber in the log. The Universal log rule is so constructed, however, that it is very easy to get from it the figures which would have been shown if any one of these other factors had been used. Assume, for example, a log 24 inches in diameter and 12 feet long, which scales by this rule 314 board feet. In order to find what the amount would have been with the factor 1.5, we have only to add a number of board feet equiv- alent to one-half the diameter, expressed in inches. Thus in this case one-half of 24 equals 12 and 314+12=326 board feet. In the same way a 12-inch log is credited by the Universal rule with 66 board feet; but if the factor were 1.5 it would scale 72 feet. This is just what the present Vermont rule allows for such a log, an amount which is considered by sawyers to be more than they can get from any but an exceptionally perfect log by careful sawing. “In order to show the precise effect of choosing different thicknesses for the allowance plank I subjoin a small table for a 34 FOREST MENSURATION. 12-foot log, and different diameters, with allowance planks of different thicknesses. Diameter in Inches, oo en 2.5 1n. 2 1n 15 1n. 1in 6 Board fee 8 11 14 17 2 rs ‘o 60 66 72 78 2 CaP pe 302 314: 326 338 36 ma o 72 742 760 778 “T am convinced that no buyer would consent to use a rule with such figures as stand in the last two columns under 1.5 inches and 1 inch, and that no seller would want to use the column under 2.5 inches. I make this comparison for the convenience of readers and in order to give the fullest opportunity for candid criticism. First-class logs carefully sawed would probably yield the amounts scheduled under the 1.5 inches caption; but the fairness and wisdom of such a rule for general use on the generality of logs is another matter.” * : : 3 : * Prof. A. L. Daniels has shown in Bulletin 102 of the Vermont Agricul- tural Experiment Station how any given rule may be expressed by a formula as follows: ‘“‘ For logs of the same length the total volume varies as the square of the diameter, and the trimmings, for reasons mentioned above, as the first power of the diameter. ‘The amount of board feet, therefore, is a quadratic func- tion of the diameter, a function of the form B=aD*?+bD +c. ‘““When now the proper values are assigned to the constants a, b, and c, the scale can be computed by arithmetic. These volumes are easily determined when the printed scale is given, as follows: Take, for example, from Doyle’s tule the amounts for a 12-foot log corresponding to the diameters 10 inches, 20 inches, 30 inches, successively, and we have: 27=100a+10b+¢, 192= 400a + 20b + ¢, 507 = 900a + 30) + ¢, from which we can easily deduce the values a=0.75, b= —6, c= 12, and Doyle’s formula reads: B=0.75D?—6D + 12. , DETAILED DISCUSSION OF LOG RULES. 35 21. The International Log Rule.—'This log rule is the result of recent investigations by Professor Judson F. Clark, Forester of the Department of Lands and Forests, Ontario. It is designed for use with a band-saw, cutting a saw-kerf of } inch. The rule is based on the formula: B/=o.22D?—0.71D, in which the first term, 0.22D%, represents the contents in board feet of a log 4 feet long, after deducting the loss in saw-kerf and shrinkage in season- ing, and the second term, 0.71D, is the waste due to square edging and to normal crook. The principles underlying the derivation of this formula are as follows: 1. An allowance of } inch is made for saw-kerf, and % inch for shrinkage and unevenness in sawing. After deducting for Gt 7 this loss, the contents in board feet of a 12-foot log are 108 See 0.66D?; and for a 4-foot log, 0.22D?. 2. The minimum board is 3 inches in width, containing not over 2 board feet. A 3-inch board must, then, be at least 8 feet long to be included; a 4-inch board, 6 feet long; a 5-inch board, 5 feet long; a 6-inch board, 4 feet long. 3. An allowance is made for a taper of 4 inch for each 8 feet of length. Professor Clark has shown that this is a conservative allowance for merchantable logs of all species so far studied in this country, including white pine, loblolly pine, spruce, balsam- fir, chestnut, and northern hardwoods. 4. Provision is made for the loss due to normal crook and that due to human and mechanical imperfections. By normal crook ‘“In the same manner we obtain the others, which are subjoined. - NCTA cos oe ve a Ss B=0.62832D?—2D. ek oon a's sin cacin:col ane B=0.75D*—6D +12. SURENOTTN So ease. vt ves Bo 50)". Memwekiatnpsiite.. 14)... <0 eee B=0.41D*?—o.1D +1. Aaa ek tts 5): 2 «005s B=0.61D?—1.7D—6. SEGRE... Sed ails = n'a dh spot B=0.62D*—1.1D—1. Potland>or Maine: .. 3.2. 0. Mane. B=0.635D*—1.45D +2. Drew ot Puget Sound. ........... B=0.615D?—4.125D +209. PHARTSRIEC SC sata su ov as en Smeeeeree B=0.555D?—0.55D —23.” FOREST MENSURATION. ol c6 06 cg CL ol c9 09 cs oS cY ov c¢ of ol 6 L ol C9 09 cs oS oS Cr ob ce of of Cz 6 8 09 cs oS Ch cr ob c¢ ce of cz Cz oz oz 8 L cr oF ce ce of of Cz Cz oz oz CI Cr C1 J 9 of of cz cz oz oz oz C1 C1 CI Ol OI Ol 9 ¢ oz se Cr C1 CI oI ol OI oI ¢ ¢ ¢ ¢ ¢ 14 oI oI ol oI ¢ ¢ ¢ ¢ ¢ ¢ CY Se See eae b g ¢ c ¢ ¢ oe eee be abd we oe OES 6 Seite he © 0 6 0 6 Ee 8k oe we hee 6 eee Se ee ew etc sees ee eee Ones ene ¢ waypuy | | (Oe 61 8ST At 9T ct FT et ot i oT 6 8 sequel t949 | 1949 ‘qaaq ul 3o7T jo yywuUeT “eae -wreIq | ‘j4oy YOut-$ Burz{No smes IOJ a[VOS PIepUeIS ‘[eouy yooy tod your $ ‘aoueMOTIe jode Ty, “ATAU DOT IVNOILVNAYALNI *suOTJOVS JOOJ-b Ij dtL'o—(z2z"0X 2q) BNO HHL is 09 ogg oggor SLbee GLze ¢Lot OSgz oggz Sgtz 6S oS Le goce ogre Coil ¢L6z og lz o06Sz ootz gs oz9t Core Crze ogo olgz Cggz CoSz ozfz LS Core cree Cee Co6z oLlz C6Sz Cr1bz obzz 9S CLE’ ooze Czoft oS gz CLoz ooSz offz OgIz c¢ o$z got C16z CrLz ogsSz o1tz CVvzz Ogoz w ts cere ol6z O1gz Stoz Cgtz Czlz Coz Cooz Q) ¢¢ Cro¢ oggz Colz CrSz o6£z CLzz Cgoz of 61 S z Co06z oS Lz 0097 oS tz oof z OS 1z Cooz CSegr & 1¢ ob Lz Crgz oo$z Ccez O1zz oLoz CZ61 Coli S of oggz otSz ootz Cgzz Cz1z C61 oS gr Crdt NS 6+ CLSz ottz Cotz oliz otoz Coor i a Cror ce gt oltz offz Olzz Cgoz CC61 ofl oolt GLS1 o LY Cofz obzz OZ IZ C661 oLgi oS Li ofgI Orc] > ot Cgzz Criz of oz O161 o6LI GLoI ogS 1 Crtr ie) a — — as i —————— ————————_ a cr Sgiz | oSoz | of61 Szg1 | Sr1Z1 | oog1t | o6b1 | og < th oLloz og61 CCer CVLI CLgI ofS1 Ccti Ozf1 OU Cv CL61 olgI LLi C9gI oogS I ogti ogf I OQZI 2 zt Cegi gli Cggl Cgcr o6t1 o6f 1 C6z1 OOTI Q It COLI Li CogI orc! Citi czrex OfzI OFIr A —— = ee a = ——— ———— a Se S| [ — oF Coli C1gI CzS1 Cev1 Cre OgzI OLII Cgor = 6¢ ozg1 Ces1 oStI COLI OgTI COrII OIIl ofol i. gt cecr CCri CLEI C6z1 OIZI Cer SSoI ¢L6 Q Le Cori ogt I oof I CzzI oS II CLor OOO! Cz6 a gt GLEI Cott ofzI OgII Sgor C1oI C+6 ¢lg ce oof I OfZI OgII C601 CzoI cc6 068 zg ve Czz1 ogtII C601 oLoI C96 006 otg SLL e¢ Of 11 o6o01 ofol oLl6 C06 oSg o6L of L ze Ogol Czo1 C96 o16 oSg C6L otl C89 1¢ C1or 096 Co6 og 00g Crl C69 otg OIS!I ogt i oIti ogf 1 Crier Cgz1 O7Z1 OLII CzI11 Cgor oto1 OOO! CC6 C16 38 FOREST MENSURATION. is meant the average crook of first-class logs accepted at the average mill. The average crook allowed in the rule is about 1.5 inches, and does not exceed 4 inches in 12 feet. Any crook more than 4 inches would have to be specially discounted by the scaler. Professor Clark first estimated this loss theoretically and then proved his computations by extensive tests at the mill. His studies showed that the waste due to crooks and surface imperfections is, like the waste in square-edging, directly pro- portional to the diameter of the log. The necessary allowance for waste in edging, crooks, etc., amounts altogether to 2.12D for 12-foot logs, or 0.71D for 4-foot logs. This allowance was determined by mathematical com- putations and by tests at the mill. With these principles estab- lished, the log table was compiled by first computing the contents of logs 4 feet long and then of logs of other lengths, allowing a taper of 1 inch in 8 feet. The International rule may be applied in mills which use saws cutting kerfs of other widths than } inch, by adding to or subtracting from the total scale a certain percentage, as indicated in the following table: For 2 -inch kerf add 1.3% Ota of f= OP Sut eet 6c 1 cc cc cc 9.5% 6c ts cc ¢ (a3 13.6% 66 3 ¢ c< (a3 17.4% a: x 66 6c 6 20.8% 22. The Doyle Rule.—This rule is known in some sections as the Connecticut River rule, the St. Croix rule, the Thurber rule, the Vannoy rule, the Moore and Beeman rule, Ontario rule, and the Scribnerrule. It is often called the Scribner rule because it is now printed in Scribner’s Lumber and Log Book. The Doyle rule is used throughout the entire country and is more generally employed than any other rule. It is the statute rule of Arkansas, Florida, and Louisiana. . % DETAILED DISCUSSION OF LOG RULES. 39 It is constructed by the following formula: Deduct 4 inches from the diameter of the log as an allowance for slab; square one-quarter of the remainder and multiply the result by the length of the log in feet. This formula does not explain the principle of the rule as well as that published over 50 years ago, giving the same results, namely: Deduct 4 inches from the diam- eter for slabs, then squaring the remainder, subtract } for saw- kerf and the balance will be the contents of a log 12 feet long, from which the others may be obtained by proportion. The principle is first to deduct a 2-inch slab regardless of the size of the log; then to square the diameter to obtain the number of the square inches on the end of the stick; deduct } for saw-kerf, then divide by 12. The result is the number of board feet in a log 1 foot long. If the last division by 12 be omitted, the contents of a 12-foot log will result. The important feature of the formula is that the width of slab is always uniform regardless of the size of the log. This waste allowance is altogether too small for large logs and exces- sive for small ones. The principle is, therefore, mathematically incorrect. The product of perfect logs of different sizes follows an entirely different mathematical law from that of the Doyle tule. It 1s astonishing that this incorrect rule which gives ridicu- lous results for very large and very small logs, should have such a general use. | Without doubt the rule has happened to yield fairly accurate results where the loss by defects in the timber and waste in milling have accidentally about balanced the inaccuracies of the rule. Generally millmen recognize the failings of the rule and make corrections to meet their special conditions. The opinions of a number of millmen regarding the rule are pertinent at this point. The following are selected from letters from a very large number of men all over the country. It should be added that the reason for such wide differences in the per- centage of inaccuracy is on account of differences in local defects of the timber sawed. 40 FOREST MENSURATION. “The saw bill will overrun the scale when the log is 20 inches and under in diameter and will begin to fall short of the scale when it reaches 24 inches in diameter, and the larger the log the more it falls short.’’—Mill located in Ohio. “Large logs have to be very straight and good to hold out, and according to our experience Doyle’s Rule might be increased 10% to 16 or 17 inches diameter, and from that up to 24 inches remain as it is and about 24 inches reduce 2 to 5%.”—Mill in Indiana. “Pine, spruce, and tamarack overrun the scale 20%; maple, ash, hemlock 5%.’—Massachusetts. “Small logs overrun 20%; large logs lose 20%.”’—West Virginia. “Fir: large logs fall short of scale 5%, small overrun 10%.” —Washington. ‘Logs 12 inches and less overrun the scale about 40% for straight, clear logs. “Logs 12-20 inches overrun the scale about 15-20% for straight, clear logs. “Logs 20-24 inches overrun the scale about 10-20% for straight, clear logs. “Logs 24-30 inches overrun the scale about 5-10% for straight, clear logs. “Logs 30-36 inches about hold up the scale. “Togs 36 and over fall below the scale.”—Mill in Ohio. “Poplar, chestnut, spruce, hemlock hold out. Cherry under- runs about 5%. Sycamore underruns 124%. White oak cut into car stock overruns 12%. Red oak cut into lumber under- runs 5%. Rock oak underruns 10%.”—West Virginia. The Doyle rule may be found in: Scribner’s Lumber and Log Book. G. W. Fisher, Rochester, way. The Woodsman’s Handbook, by Henry S. Graves, Bull. No. 36, Bureau of Forestry, Washington, D. C. The instructions given in Scribner’s Lumber and Log Book are to measure the log at the middle. In practice the logs are measured at the small end inside the bark, except long logs, which are measured at the middle. One of the writer’s corre- spondents measures the diameter of a long log at one-third the » DETAILED DISCUSSION OF LOG RULES. 41 distance from the small end. Long logs are those containing two or more short logs of merchantable length. 23. The Scribner Rule.—This is the oldest log rule now in general use. It was originally published in The Ready Reckoner, by J. M. Scribner. It is now usually called the “Old Scribner Rule.” It is used to some extent in nearly every state, and is the statute rule of Idaho, Minnesota, Oregon, Wisconsin, and West Virginia. The rule was based on computations derived from diagrams drawn to show the number of inch boards that can be sawed from logs of different sizes after allowing for waste. The following description of the rule is taken from the edition of 1846: “This table has been computed from accurately drawn dia- grams for each and every diameter of logs from 12 inches to 44, and the exact width of each board taken after being squared by taking off the wane edge and the contents reckoned up for every log, so that it is mathematically certain that the true con- tents are here given, and both buyer and seller of logs will un- hesitatingly adopt these tables as the standard for all future con- tracts in the purchase of sawlogs where strict honesty between party and party is taken into account. In these revised com puta- tions I have allowed a thicker slab to be taken from the larger class of logs than in the former edition, which accounts for the discrepancy between the results given in these tables and those in former editions. “The diameter is supposed to be taken at the small end, inside the bark, and in sections of 15 feet, and the fractions of an inch not taken into the measurement. This mode of measurement, which is customary, gives the buyer the advantage of the swell of the log, the gain by sawing it into scantling, or large timber, and the fractional part of am inch in the diameter. Still it must be remembered that logs are never straight and that oftentimes there are concealed defects which must be taken as an offset for the gain above mentioned. It has been my desire to furnish those who deal in lumber of any kind with a set of tables that can 42 FOREST MENSURATION. implicitly be relied upon for correctness by both buyer and seller, and to do so I have spared no pains nor expense to render them perfect; and it is to be hoped that hereafter these will be preferred to the palpably erroneous tables which have hitherto been in use. If there is any truth in mathematics or dependence to be placed in ihe estimates given by a diagram, there cannot remain a particle of doubt of the accuracy of the results here given.” The judgment of most sawyers with whom the author has talked is that the Scribner rule gives very fair results for small logs, but that for large logs, for example those above 28 inches, the results are too small if the logs are free of defects. It often hap- pens that defects are greater with large logs than with small ones, because the former are from trees which are older and more apt to be mature or overmature than small trees. Scribner’s tule is fairly satisfactory in such cases. The results for the small sound logs are fairly accurate and the defects of the larger logs is balanced by the deficiencies of the rule. Sometimes the Scribner rule is converted into what is known as the Scribner Decimal rule by dropping the units and rounding the values to the nearest tens. Thus 107 board feet would be written 11 in the decimal rule; 104 would be written 10. The Hyslop rule is practically the same as the Scribner Decimal rule. The original rule did not give values below 12 inches. A number of lumber companies have interpolated for their own use the values for small logs. ‘Thus the figures for small diam- eters in the table on page 24 are those used by a company in the Adirondacks. The Lufkin Rule Company publishes three tables for the scale of small logs by the Scribner rule. These are called Decimal A, B, and C. , The rule is now published in the following books: Scribner’s Rule for Log Measurement. George W. Fisher, publisher, Rochester, N. Y. Price 30 cents. The Woodsman’s Handbook, by Henry Solon Graves. Bulletin No. 36, Bureau of Forestry, Washington, D. C. 24. The Maine Rule.—This is also known as the Holland rule, DETAILED DISCUSSION OF LOG RULES, 43 and as Fabian’s rule. Its use is restricted to northern New Eng- land, chiefly to Maine, where it has long been the principal log rule. The Maine rule was constructed by the use of diagrams representing the small ends of logs of all diameters from 6 to 48 inches. ‘The inscribed square of the logs was first determined, and the contents of the logs were then computed by allowing t inch for each board and one-fourth of an inch between the boards for saw-kerf. ‘The boards outside the square were reck- oned, if not less than 6 inches in width; otherwise the whole slab was discarded. In judging this rule it must be remembered that it was devised for the measurement of short logs and not for long logs, to which it is now so frequently applied. Millmen very generally agree that the Maine rule is extremely satisfactory for short logs. In fact, it probably comes nearer satisfying the present require- ments of a modern sawmill than any of the other rules in com- mon use. It gives considerably larger results than the Scribner, Hanna, and Spaulding rules. Like the Scribner rule the values in the Maine rule run irregularly. It is a very simple matter to correct these irreg- ularities by graphical interpolation. Mr. H. D. Tiemann has prepared a corrected form of the Maine rule as given in the table on page 44. The chief trouble with the rule is not due so much to defects in the values as to the present method of applying it. As explained in section 37, logs over 30 feet are measured as two logs, the diameter at the small end being measured and the diameter at the middle being estimated. An illustration of the working of the rule is given by Mr. Austin Cary in the Report of the Forest Commission of Maine, 1896: “Even when the present method of scaling by the Maine rule is conscientiously applied it leads to wasteful lumbering. A good example is that of a spruce log, cut in Maine, 32 feet long and scaled as 169 board feet by the Maine rule. If the loggers had cut this particular log 4o feet long the scale would have 44 FOREST MENSURATION. MAINE RULE. (Values Made Regular by Interpolation.) Length in Feet. Diameter, 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 Inches. Board Feet. 6 13 15 17 20 22 25 27 2 7 19 2 26 30 34 38 2 45 8 26 31 36 42 47 52 58 63 9 34 41 47 54 61 68 75 82 10 43 51 59 | 68 7 85 93 102 II 2 62 73 83 93 104 114 12 12 62 75 87 99 112 124 136 149 13 73 88 102 117 132 146 161 176 14 85 102 119 136 153 170 187 204 15 98 118 537 157 176 196 216 235 16 112 135 157 180 202 225 247 270 17 128 154 179 205 231 256 282 307 18 145 175 204 233 262 292 321 350 19 164 197 230 263 296 329 362 395 20 184 221 258 295 332 369 406 443 21 205 247 288 329 370 411 452 494 22 228 274 319 364 410 455 501 547 23 252 302 352 403 453 503 554 604 24 277 332 387 443 498 553 609 664 | been only 168 board feet, or 1 foot less, although 8 feet longer and containing 5 cubic feet more. The contractor gained by wasting 8 feet of wood entirely suitable for saw lumber or pulp. Other facts about this tree are shown in the table below and illustrate the tremendous range of results depending on the dif- ferent methods of applying a single specified rule: Log as cut, 32 feet long; contents with bark, 31 cubic feet; scaled as two 16-foot logs, giving butt log 1 inch rise................... 169 ft. Scaled as two 16-foot logs, giving butt log actual rise............ 135°" A 16-foot log above, that might have been taken, scales. ......... 3x. { Log if cut 40 feet long; contents with bark, 36 cubic feet; scaled as two 20-foot logs, giving butt log 2 inches rise..........-...... 168 ‘‘ Scaled as two 20-foot logs, giving butt log actual rise............ 1985 Whole tree, sawed down at 1} feet from ground and taken up to 6 inches diameter; contents, 43 cubic feet; scaled in 16-foot lengths from butt up with actual diameter. .................. 226% Total contents of the tree’s stem, 46 cubic feet.’ Scaled as four 10-foot logs, using actual top diameter of each...... 21q°>* ; d a ; DETAILED DISCUSSION OF LOG RULES. 45 This illustration is sufficient to show what absurd results may be obtained if the rule is loosely applied, as is customary in prac- tice. The Maine rule may be obtained from V. Fabian, Milo Junction, Maine. 25. The Hanna Rule is used in Pennsylvania, Tennessee, Vir- ginia, New York, and Massachusetts, and locally elsewhere. It was computed from diagrams drawn for every size of log from 8 to 50 inches in diameter. Practical millmen were consulted by the author of the rule to check the results. This rule was con- structed on the same general principles as the Scribner, Maine, and Spaulding rules and its results, like those rules, are fairly satisfactory for logs of average size. The use of the rule among practical millmen is on the increase. The rule follows closely the Scribner rule, but is less erratic. When the two are compared the Hanna rule appears like an attempt to correct the eccen- tricities of the Scribner rule. . 26. The Spaulding Rule.—This is the statute rule of Cali- fornia adopted by an act of the Legislature in 1878. The revised statute is quoted in the Appendix. This rule was constructed from diagrams. Its author has given no adequate description of how the results were obtained. The only description of the construction of the rule as published by its author is as follows: | “Each sized log has been scaled so as to make all that can be practically sawed out of it, if economically sawed. Each log to be measured at the top or small end, inside of the bark, and if not round, to be measured two ways—at right angles—and the average taken for the diameter. Where there are any known defects, the amount to be deducted should be agreed upon by the buyer and the seller, and no fractions of an inch to be taken into the measurement.” “In the foregoing table I have varied the size of the slab in proportion to the size of the log, and have arranged it more par- ticularly for large logs by taking them in sections of 12 feet and 46 FOREST MENSURATION. . carrying the table up to 96 inches in diameter. As there has never been any in use for scaling over 44 inches, it has been my purpose to furnish a table for the measuring of logs that can be implicitly relied upon for correctness by both the buyer and seller; and to do so, I have spared no pains or expense to render it perfect.” The rule gives very fair results for logs which are sound. Where the run of logs is defective, the scale overruns the saw bill. As far as the author can learn, the millmen are satisfied with the rule. It may be obtained from N. W. Spaulding Saw Company, 18 Freemont Street, San Francisco, California. 27. The British Columbia Rule.—The province of British Columbia has established a statute log rule which is based on the following mathematical formula: For logs up to 4o feet in length deduct 14 inches from the diameter at the small er'd inside the bark; square the result and multiply by the decimal 0.7854; from the product deduct three-elevenths; multiply the remainder by the length of the log and divide by 12.. For logs over 4o feet ‘in length an allowance is made on half the length of the log in order to compensate for the increase in diameter. This allow- ance consists of an increase in the diameter at the small end of the log of 1 inch for each 1o feet in length over 4o feet. Thus for logs 51 to 60 feet long the contents of half the log are com- puted by the diameter at the small end. The other half is con- sidered to have a diameter 1 inch larger. The British Columbia log table is published in a small booklet entitled The British Columbia Log Scale, and may be secured in Victoria, B.C., at the cost of $2.00. 28. The Drew Rule.—This rule has been adopted as a statute rule in the State of Washington. The law establishing it is given in full in the Appendix.. It is also the official log rule of the Puget Sound Timbermen’s Association. It is used only in the north- western states and is confined chiefly to Washington. No infor- mation is given in the published rule as to how the values were = 72 EEE DETAILED DISCUSSION OF LOG RULES. 47 derived. The rule may be obtained in the form of a small leather- bound booklet for $2.50 in Seattle. 29. The Constantine Rule.—This is a theoretical rule, which gives the solid contents, expressed in board feet, of logs without any reduction for waste in sawdust, slabbing, edging, etc. The principle of the rule is to determine, first, the solid contents, in cubic feet, of cylinders of different diameters. ‘These results are then translated into board feet by multiplying by 12, on the principle that if there is no waste in sawing, there will be in each cubic foot 12 board feet. Of course, it is really not proper to express the solid contents of a log in board feet, because the board foot represents a manufactured preduct, and it is impos- sible from a cubic foot of round timber to saw 12 board feet. As it stands, therefore, the Constantine rule can hardly be considered a log rule, but only a mathematical step in the derivation of a log rule. The formula for constructing a table for this rule is as follows: Square the diameter of the small end of the log inside the bark and multiply by the decimal 0.785; multiply the result by the length of log and divide by 12. Expressed algebraically, D Bi= [e xX L+ 144 | Ts, Bj =0.7854D? X L~+12. A practical board rule is sometimes made from the Constan- tine table by deducting a third or fourth from the figures for saw-kerf and other waste. This rule is also used as a founda- tion for the Champlain and Universal rules of Professor Daniels. 30. The Canadian Rules.—A number of the American log rules are used in Canada, but those recognized by statute or custom are the Quebec rule, the New Brunswick rule, the British Columbia rule, and the so-called Ontario rule, which is the same as the Doyle rule. The British Columbia rule has already 48 FOREST MENSURATION. been discussed in connection with the Pac#ic Coast rules. The author has been unable to learn exactly how the Quebec and New Brunswick rules were constructed. As far as the author is informed, the Canadian rules, except, of course, the Doyle rule, have never been used in this country. 31. Miscellaneous Log Rules. The construction of the Vermont Rule is described in the Appendix. It still has some use in the northeastern states, being retained chiefly because it is a State rule. An effort is now being made to replace it by a new rule, it being considered unsatisfactory by most lumbermen. It is also called the Humphrey rule and the Winder rule. The Baughman Rules.—These rules, devised by H. R. A. Baughman, of Indianapolis, have been constructed to show the full contents, in board measure, of perfect logs manufactured by modern machinery; one to be used with a rotary saw, and the other with a band saw. They are based on diagrams. As in other diagram rules, the values increase by irregular differ- ences. It is, however, one of the few rules which do not allow an excessive amount for waste. Its values are, for small logs, almost identical with those of the Champlain rule. The Baugh- man rules are contained in Baughman’s Buyer and Seller, Indian- apolis, Ind., 1905. Price $1.50. The Baxter rule is used chiefly in Pennsylvania. According to Prof. J. T. Clark it is based on the formula: Subtract 1 from the diameter inside the bark at the small end, square the re- mainder and multiply by 0.52, and the result is the contents of a 12-foot log. The Dusenberry Rule is used in Pennsylvania, New York, Ohio, and Indiana. It was made originally for white pine, but is now used also for other timbers. The Favorite, or Lumberman’s Favorite Rule, is used in Vir- ginia, West Virginia, Michigan, New York, Texas, Tennessee, Indiana, Pennsylvania, North Carolina, and Missouri. It is ‘probably based upon diagrams. The Square of Two-thirds Rule, which is also known as the St. a DETAILED DISCUSSION OF LOG RULES. 49 Louis Hardwood rule, the Two-thirds rule, the Tennessee River rule, and the Lehigh rule, is used in Tennessee, Pennsylvania, North Carolina, Kentucky, Illinois, Indiana, New Jersey, Virginia, and West Virginia, and probably in some other states. It is based on the following formula: Deduct one-third of the diameter at the small end of the log inside the bark for saw-kerf and slab, square the remainder, multiply by the length, and divide this product by 12. .The result is the contents in board feet. The Square of Three-jourths Rule, whose other names are the Portland scale, the Noble and Cooley rule, the Cook rule, the Crooked River rule, and the Lumberman’s scale, is occasionally used in the northeastern states. The formula upon which it is based is as follows: Deduct one-fourth of the diameter at the small end of the log inside the bark for saw-kerf and slab, square the remainder, multiply by the length of the log, and divide this last product by 12 for the contents in board feet. The Cumberland River Rule is known in some parts of the country as the Evansville rule and the Third and Fifth rule. It is used in Tennessee, Kentucky, Indiana, Ohio, Michigan, Illinois, Massachusetts, and probably in some other states. It is based upon the following formula: Deduct one-third from the daimeter at the small end inside the bark to reduce the round log to square timber. Then from one side of the square thus obtained deduct one-fifth for saw-kerf; multiply the remain- der by the side of the square and the product will be the contents of a log 12 feet long. For logs of other lengths multiply by the length and divide by 12. This rule was constructed for the measurement of hardwood logs in the water in the Mississippi River and its tributaries. These logs are often defective and in the water it is impossible to distinguish the defects which are hidden by the water itself, by mud, sand, plugs, etc. The log rule is supposed to allow for all such hidden defects. The Herring Rule, which is also called the Beaumont rule, is used in Texas. It was first published in 1871 by T. F. Herring, Beaumont, Texas, and afterwards enlarged by W. A. Cushman 50° FOREST MENSURATION. of Beaumont. No intimation is given of exactly how the table was constructed. It is probable that it is based partly on the actual cut of the logs at the mill and partly on diagrams. The rule may be obtained from Mark Weiss, Beaumont, Texas. Price $1.00. The Orange River Rule is also known as the Ochiltree rule and as the Sabine River rule. It is used in Texas. It is based on the following formula: Multiply the square of the diameter of the small end of the log inside the bark by the length of the log and divide the product by 30; the result is the contents in board feet. The Combined Doyle and Scribner Rule.—This is a combina- tion of the Scribner and Doyle Rules. It is used in New York, New Jersey, Pennsylvania, Virginia, Tennessee, Kentucky, Ala- bama, Lovisiana, Arkansas, Mississippi, Missouri, Indiana, IlIli- nois, Michigan, Ohio, Iowa, Wisconsin, Montana, Idaho, South Dakota, and probably elsewhere. It has been adopted as the official scale of the National Hardwood Lumber Association, St. Louis, and is published in their “Grading Book.” The values for diameters under 28 inches are taken from the Doyle rule; those for 28 inches and over from the Scribner rule. The effort seems to have been to find a rule which gives very small results, in order to cover loss in defective timber. The principal use is with the hardwocds, which are apt to be unsound. It is to be counted as one of the rules designed for a special class of timber. The Chapin Rule is based on measurements of logs actually sawed into lumber. It is claimed that it gives the greatest amount of lumber which can be manufactured from straight smooth logs. It is a comparatively new rule and has not yet come into very general use. It may be purchased from the American Lumber- man, Chicago. A number of other rules are in local use. They are as follows: The Northwestern Rule, used to some extent in Michigan and Illinois. DETAILED DISCUSSION OF LOG RULES. 51 The Derby Rule, also known as the Holden and Robinson rule, which is used in Massachusetts. The Partridge Rule, also called the Murdoch and Fairbank rule, which is used rarely in Massachusetts and which is based on ¢ inch boards. The Preston Rule. ‘This is based upon the principle that one- fifth of the contents of a log should be deducted for saw-kerf. The waste in slabs is calculated by deducting 13 inches for small logs and 14 inches for large logs. The results are given in board feet and inches. The Parsons Rule, used in a few places in Maine. The Ropp Rule, used in Illinois. It is based on the following formula: Subtract 60 from the square of the diameter of the small end of the log inside the bark, multiply the remainder by half the length of the log, and point off the right-hand figure. The Stillwell Rule, known as the Stillwell Vade Mecum rule, used by its author in Georgia. The Saco River Rule, used in Maine. The Wilson Rule, used in Massachusetts. The Ballon Rule, used by M. E. Ballon & Son, of Becket, Mass., in measuring small hardwood timber, such as_ basket ash. The Wilcox Rule, used locally in Pennsylvania for softwood timber. The Warner Rule, used locally in New York. The Boynton Rule, based upon a compromise of the Vermont and the Scribner rules and adjusted by sawyers’ tallies. It is used in Vermont. The Carey Rule, used in Massachusetts. The Forty-five Rule, used in New York. It is based upon the following rule: For a 24-inch log multiply the square of the diameter, namely 24, by the length of the log and the result by 45, then point off three places. The figures at the left of the decimal point will represent the contents in board feet. For every variation of 2 inches in the diameter from the standard 52 FOREST MENSURATION. 24-inch log add or subtract 1 from the number 45 in the formula, according as the diameter is larger or smaller than 24 inches. The White Rule, used to a limited extent in Montana. The Finch and Apgar Rule, published in the Excelsior Log Book Table, New York. The Ake Rule, used locally in Clearfield County, Pa. It is based upon the following rule of thumb: Multiply the diameter of the log, measured at the small end inside the bark, by 0.7; square the result; multiply the product by the length of the log and divide by 12. The final result will be the contents in board feet. The Younglove Rule is a very old rule formerly used in New England and probably still occasionally employed in Massa- chusetts. Other rules may be in existence, but they are unknown to the author. CHAPTER IV. LOG RULES BASED ON STANDARDS. 32. Definition of Standard Measure.—It was shown on page 26 that the custom of using a standard log of specified dimensions as a unit of volume has been used for over fifty years. A table of standards is based on the principle that the contents of logs vary directly as their lengths and the squares of their respective diam- eters. To obtain the volume of any given log in terms of a specified standard, square the diameter at the small end and divide by the square of the diameter of the standard log; then divide by the length of the standard log and multiply by the length of the log measured. Thus if the standard is a log 12 feet long and 24 inches in diameter at the small end, the square of the diameter of the log measured is divided by the square of 24 and then multiplied by a fraction whose numerator is the length of the given log and denominator the length of the standard. Expressed algebraically, the rule for determining the volume of a log in standards is 1 Baa emer ant in which V is the volume of the log, D its diameter at the small end, L its length, d and / the diameter and length of the standard. It will be noticed that in this formula the full contents of the standard and of the log are not compared, but the contents of cylinders having diameters equal to the diameters of the respec- tive logs at the small end. If the full contents of the logs were 53 54 FOREST MENSURATION. to be compared, it would be necessary to take the measurements of diameter at the middle. ‘This will be clear by reference to the formula for determining the solid contents of logs described in section 47. 33. The Nineteen-inch Standard Rule.—One of the standards in most common use is the so-called 19-inch standard, or market. The unit is a log 13 feet long and 1g inches in diameter at the small end inside the bark. On the principle that the contents of logs vary as the squares of their diameters, a 10-inch log 13 feet long contains 0.28 standards (the square of 10 divided) by the square of 19). Expressed algebraically the formula for deter- mining the contents of a given log by the 19-inch standard rule is in which V represents the volume in standards, D the diameter inside the bark at the small end, and L the length of the log. This log rule is most commonly used in the Adirondack Mountains of New York. It is particularly popular in measuring pulp wood because the rule is based on volume and not on board measure. It is sometimes called the Glens Falls Standard rule. It has been called by some the Dimick rule because it is published in Dimick’s Ready Reckoner. This booklet, edited by L. Dimick, may be purchased for 25 cents from Crittenden and Cowles, Glens Falls, N. Y. Standard measure is commonly translated into board measure by multiplying the volume of a given log in standards by a con- stant. In the case of the Nineteen-inch Standard rule, it. is assumed that one standard is equivalent to 200 board feet, and the number of standards in a log, regardless of its size, is multiplied by 200 in translating from standard to board measure. This procedure is emphatically incorrect, because the contents of logs measured in standards vary as the squares of the diameters, while the contents of logs measured in board feet vary by a totally LOG RULES BASED ON STANDARDS. 55 different rule (see sections 19-21). When a standard table is converted into board measure by multiplying throughout by a constant, as for example 200, it inco rectly is assumed that the board contents vary as the squares of the diameters of the logs. When logs of different diameters are scaled both in standard measure and board measure, the results are not the same as when the logs are scaled in standard measure and converted into board measure by multiplying by 200. It is true that the average of a very large lot of logs when measured by the two scales will run about 200 feet to the standard (based on Doyle’s rule). This is the only way that the converting factor can correctly be used. It should not be used when applied to individual diameters. The table on page 56 shows that, taking logs separately, there are not 5 standards to the thousand, but from 4 to nearly 14 standards to the thousand, according to the diameters of the logs. \. 34. The New Hampshire Rule (Blodgett Rule). Although usually not recognized as a standard log rule, the Blodgett rule, which has been adopted as the statute rule of New Hampshire, is nothing more nor less than a standard rule based on the same principles as that of the ‘Adirondack market described above. The Blodgett standard, as fully described on page 363, assumes as a unit a log 1 foot long and 16 inches in diameter. The contents in so-called cubic feet (more correctly standards) of a log of any dimensions is found by the following formula: in which V is the volume in standards, D the diameter in inches, and L the length of the log in feet. This rule is now being very generally introduced in the spruce region of northern New England for the measurement of long logs which are cut for pulp. The reason for its popularity is because it is a volume rule. In the manufacture of wood pulp 56 FOREST MENSURATION, COMPARISON OF 13-FOOT LOGS SCALED IN STANDARDS AND BOARD MEASURE. (From Report of N. Y. Forest Commission, 1894). Pilenraken Stendard Number of Number of Number of Number of Inches.® Measure. see fe Oh Loans | Oe 8 Pe iy oe 5.6 13 76.9 13.7 9 .22 4.5 20 50.0 1G ee 10 277 3.6 29 34.5 9.6 I! 335 3.0 40 25.0 8.3 12 - 399 22% 52 19.2 Ye 13 . 468 ar 66 r5.1 742 14 - 543 1.8 81 12.3 6.8 15 .623 1.6 98 10.2 6.4 16 . 709 1.4 117 8.5 6.1 17 . 800 ‘3 137 7 ae 5.8 18 . 897 1 159 6.3 5.7 19 1.000 1.0 183 5*5 Sam 20 1.108 me) 208 4.8 ee 21 1.221 .8 235 4.2 Ler. 22 1.341 he 263 3.8 5.2 23 1.465 on 293 a4 5.0 24 1.595 .6 325 S.1 4.9 25 Rigs 6 358 2.8 4.8 26 r,o72 5 393 a5 4.7 a7 2.020 5 430 a's 4.6 28 ae © ss 468 ay 4.6 29 %. 390 ey] 508 2.0 4.6 30 2.493 -4 549 1.8 4-5 31 2.662 4 592 oer 4.5 32 2.836 Me 637 1.6 4.4 33 3.017 “3 683 mo 4.4 34 3.202 +3 731 1.4 4.3 35 3-393 3 781 1.3 4.3 36 3.590 3 832 1.2 4.3 * At top end of log, inside the bark. Tt Doyle's Rule. the entire log is utilized, there being very little waste. Land- owners are therefore demanding a unit of measure which will take into account the entire contents of the logs. Another reason for the adoption of the New Hampshire rule is the widespread dissatisfaction with the Maine rule as it is now used. The reader is referred to the discussion of the New Hampshire rule in sections 37 and 38. Just as in the case of the Adirondack standard, lumbermen are accustomed to convert the Blodgett rule into board measure. LOG RULES BASED ON STANDARDS. 57 The statute states that the ratio of the Blodgett standard to the thousand feet shall be as 100 is to 1000, or 10 feet in every cubic foot. In practice the lumbermen consider that there are 115 Blodgett feet in tooo board feet when the diameter measure- ment is taken at the middle of the log and 106 Blodgett feet per 1000 board feet when the measurement is taken at the small end of the log. These are fair average figures and in practice are applicable in converting the scale of a large lot of logs Jumped together from one measure to the other. It is not, however, fair to construct a log table for board measure by dividing the values in the Blodgett rule by the constants 106 or 115. Such a log rule for board measure still remains a volume rule, although expressed in board feet. The values in the table are not proportional to the board measure of the log, but to the cubic volume measure. 35. The Cube Rule.—Another standard rule is the so-called Cube rule of the Ohio River. This is based on the hypothesis that a log 18 inches in diameter is the smallest one from which a 12-Inch square piece can be cut. To use local phraseology, an 18-inch log will cube once, meaning that for each linear foot there will be one cube. To estimate the contents of a log, square the diameter in inches, multiply by the length in feet, and then divide by the square of 18. Algebraically, Ordinarily 12 board feet are allowed for one cube. This rule is known also as the Big Sandy Cube Rule. 36. Other Standard Rules.—The Twenty-two Inch Standard Rule is still used to some extent in New York State and probably elsewhere. The unit is a log 12 feet long and 22 inches in diam- eter at the small end inside the bark. The rule is used in the same way as the Nineteen-inch Standard rule, and a table may ' be constructed on the same principle. The 22-inch standard log “"T9 _ ~——— =_ 14 SCRISNER'ST ) ee FOREST MENSURATION. ta =~ 13_—SPAULDING 62 2 —==__ 58 eetteita 1 AN — aes 12—scaALe ~=1 ~=16= soripNER =! (ns = 12-——SCALE 12} 19 37/=48| =59! 73|_— = = __24—=DECIMAL =1 —_=22 1 Aa | | — an SSOALE. 1 _——— a —S—povte — =. 12 SCRIBNER 43 SCRIBNER 2 iFIc. 1.—Different Forms of Scale Rules. 5 10 = omen LOG RULES BASED ON STANDARDS. 59 contains 252 board feet (Scribner rule). Common usage gives four standards to the thousand board feet. The Twenty-two Inch Standard rule is sometimes called the Saranac River Standard rule. The ‘Twenty-four Inch Standard rule is based on a standard log 24 inches in diameter inside the bark at the small end and 12 feet long. ‘The standard log contains 300 feet, board measure, according to the Doyle Rule. In the use of this rule timber is usually sold by the standard or by the 300 feet, instead of by the thousand feet, as commonly; the logs are scaled by the Doyle rule and the total number of feet divided by 300, the unit of sale being a certain sum per standard. To obtain the value of the odd number of. feet, the latter are divided by 300 and multiplied by the price per standard. The Canadian standard rules are based on logs 12 feet instead of 13 feet in length, and 21 and 22 inches, respectively, in diameter. These rules are used in the same way as the Amer- ican standard rules already described. CHAPTER V. METHODS OF SCALING LOGS. 37. Instruments for Scaling Logs.—The measurement of logs to ascertain their contents is called scaling. The instrument used for measuring logs is called a scale stick, scale rule, or log rule. A number of different types are manufactured. ‘The most com- mon type of scale rule consists of a stick, sqnare or flat, which may be placed on the end of a log and shows, by two sets of figures on its face, both the diameter of the log and its contents in board feet. At each inch-mark is indicated the volume in board feet, by a specified rule, of a log of that diameter. Each line of figures represents the results for one length of log, the lengths being indicated at the left-hand end of the stick. It is exactly as if a printed log rule were wrapped about the stick. The flat type of stick is most commonly used throughout the country. These rules are generally made of. hickory and tipped with a plain binding of brass or by a head of iron. There are in use a variety of such heads for the measurement of logs of different forms. Fig. 1 shows a number of forms made by the Lufkin Rule Co., Saginaw, Mich. The ad- vantage of a head is that the rule may be placed quickly and accu- rately on the end of the log. If there is no such guide, inaccuracies are frequent through carelessness in not placing the end of the rule exactly at the edge. This type of rule, however, is applicable only where the log has been peeled. Where logs are scaled with the bark on, the plain rule with no guide-head must be used, or a reduction in the measure made for the thickness of the bark. 60 METHODS OF SCALING LOGS. 61 Sometimes logs are “nosed’’; that is, the sharp edges are rounded off with the axe to prevent splitting (‘‘brooming”’) of the ends in transportation. In scaling such logs a long guide-head on the scale-stick is needed. Occasionally scale-sticks are made hexagonal instead of flat or square. The old Cary and Parsons . scales of Maine were formerly constructed in this way. Where scaling in the woods consists merely in measuring the diameters of the logs, a flat rule graduated in inches and half inches is used. These rules are often made by the scalers themselves, or for them by the camp blacksmith. A common type consists of a flat steel rule 1 inch wide attached to a wcoden handle. | Several firms manufacture a caliper scale for the Scribner rule. Calipers are -used also where the diameter is meas- ured at the middle or at one-third from the end. The New Hampshire rule requires a measurement at the middle of the log. Therefore a caliper rule is used. The most common form is one in which there_is a depression on the inside of each arm, so that the recorded diameter is less than the real diameter. This is the allowance for bark. These calipers are constructed for use with spruce, and an allowance is made on the calipers equivalent to the average thickness of spruce bark at the middle of an average log. The scaler is thus saved the trouble of chipping off the bark or of measuring its thickness. It is, of course, a rough method to assume that on all logs the thickness of bark is the same. For measuring the lengths of logs a wheel is often used. It consists of ten spokes, each tipped with a spike, mounted on a small hub which is attached to the caliper scale. The spokes are all painted black except one, which is yellow, and this one is weighted with a band of lead, so that it always points down- ward when at rest. When the wheel is placed on a log, the yellow spoke touches the log first. The construction is such that the tips of the spokes are 6 inches apart. When the wheel is run along a log, each revolution, easily counted by the yellow spoke, measures 5 feet, and as the distance between the spokes is 6 inches, 62 FOREST MENSURATION. the length of a log may quickly be determined to within 6 inches, (Fig. 2.) Fic. 2.—Wheel for Measuring Lengths of Logs. 38. Methods of Measuring the Diameters and Lengths.— The | methods of scaling logs differ in using different rules and accord- ing to local differences in the character of timber, in the market requirements, in the habit of the individual scalers, etc. In regions where the logs are cut into short lengths and piled on skidways for winter hauling, as in the Adirondacks, the scaling is done in the following way: Ordinarily two men constitute the scaling crew. They are provided with a rule for measuring the diameters of the logs, a note-book, tally-sheets or a“ scale-paddle”’ for recording the measurements, a special marking-hammer, and crayons for marking the logs. One scaler measures the | diameters of the logs inside the bark at the small end; the other records the results. Only the smallest diameter is recorded, since the log tables are based on length and on diameter at the small end of the log. It is not necessary to measure separately the length of each log, for there are usually only a few standard lengths, as, for example, 10, 12, 13, 14, and 16 feet. The scaler can tell at a glance the correct length. If a log is slightly longer than the standard, the extra length is disregarded. For example, METHODS OF SCALING LOGS. 63 a log 16.5 feet long is scaled as a 16-foot log. If 18 feet is the next standard length, a log 17.5 feet long is scaled as a 16-foot log. ‘Therefore, a log may be slightly longer than the specified length but never shorter. If a log is shorter than the length of the shortest specification (ordinarily 8 or ro feet) it is discarded entirely. A great deal of waste is caused by choppers through careless measurement of log lengths. In measuring the ends of logs, the diameters are rounded to: whole inches. If a diameter is nearer 7 than 6 inches, the log is tallied as 7 inches. If the diameter is exactly between two. whole inches, as, for example, 9.5 inches, the scaler usually tallies it under the lower inch class, in this case 9. Sometimes scalers. endeavor to throw about half of such logs into the inch class below and half into the class above. Very conservative scalers record all diameters falling between two whole inches in, the lower inch class, even if it is within one-tenth of an inch of the next class (for example 6.9 inches would be called 6-inches). When logs are evidently not round, the rule is usually placed at a point on the cross-section where the diameter is about an average between the largest and smallest dimensions. Some scalers always take the smallest diameters, a precaution necessary in measuring veneer logs. The field records are taken on special forms prepared by the company owning or buying the logs. Often the scalers use a blank-book or wooden scale-paddle in the woods, and then trans- fer the figures to regular forms at the camp. There are two methods of recording the measurements. The most common way is to tally the logs by diameter and length, and then afterwards compute the volume in the office. The other way is to record the board contents of each log as shown by the scale-stick. When a log has been scaled, the end is chalked to prevent its measurement a second time. Logs which are to be discarded receive a special chalk-mark. At this time or later the logs are stamped with the special marking-hammer of the purchaser of 64 FOREST MENSURATION. the logs. It is customary in many places to blaze a tree near each skidway, and mark the number of the skidway and number of logs tallied. Thus 7%; would mean that there are 460 logs on skidway number 23. The description of scaling given in the previous pages applies to the northern regions where logs are cut short and where roads lare used for hauling. The principles of scaling are practically the same in other sections where short logs are cut. When the logs are loaded on cars in the woods, the scaling is generally done on the cars after loading. Where logs are to be driven, they may be scaled on the bank before rolling into the river, or, where slides are used, at the side of the slide before they are started. Naturally the accuracy of the different scalers varies tremen- dously. Some guess at the dimensions of many of the logs with- out measuring them, and even estimate the total run of a pile without bothering to measure any of the logs in it. In Maine and also in some parts of New Hampshire, spruce is cut in long logs, that is, the entire merchantable part of the tree is taken out in one log. The scaling is sometimes done as the logs are hauled to the skidways or yards, and sometimes at the landing if they are to be driven. If the Maine Log rule is used, the scaler’s outfit consists of the ordinary Maine scale- stick, a measuring-pole or tape, marking-hammer, and chalk and note-book. The small end of the log and its length are measured. The results in board feet are read directly from the stick and recorded on special tally-blanks or in a note-book. The Maine rule gives figures for lengths only up to 30 feet, so that if a log is longer than that, it must be scaled as two logs. Ordinarily the diameter at the small end alone is measured, the scaler estimating the diameter at the middle. Thus if a log is 36 feet long, the small diameter 7 inches, and the diameter at the center estimated at 9 inches, the contents of two 18-foot logs, respectively g and 7 inches in diameter, are read from the stick as the contents of the whole log. The scaler guesses at the middle diameter of the log after measuring the top. The increase in size from top to center (called the “‘rise”) may be estimated METHODS OF SCALING LOGS. 65 very accurately by experienced scalers. Sometimes a scale-stick is used which gives the contents of whole logs over 28 feet long, constructed on the principle that logs 28 to 32 feet long have a rise from tip to center of 1 inch, those 32 to 36 feet long a rise of 2 inches, those 36 to 4o feet long a rise of 3 inches. The rise of logs over 40 feet long is left to the scaler’s judgment. The stick thus constructed is called the regular five-line rule. Deductions for crooks and other defects are made according to the judgment of the scaler. There are no rules, the discounting being entirely a matter of experience. In common practice it is mostly customary to reduce the total scale of a lot of logs by a certain percentage as a factor of safety. This is particularly the case where the quality of logs is extremely poor. For example, the disease of cypress called “peckiness” is so difficult to dis- cover from external signs that a general reduction for safety is necessary. The growth of the pulp industry in Maine has introduced a new factor in the scaling of spruce. Inasmuch as the whole log is used in making pulp, a solid measure is more appropriate than board measure. For this reason many operators are now using the Blodgett rule. This requires the measurement of the middle diameter of a log instead of the end diameter. The measurement is taken with calipers of the type described before. The length of the log is measured and the middle point located by a wheel. The diameter is taken outside the bark, the calipers being con- structed to allow for an average bark width. The contents of the log are read directly from the beam of the caliper. The deduction for defects is made as with the Maine rule. In scaling long logs by the Doyle rule, the diameter is measured at the middle or the two ends are averaged. Better results are obtained if long logs are measured in short lengths and the diameters taken at the points where the cuts would be made. 39. Methods of Making Discount for Defects.—If all the logs on a skidway were sound and straight the operation of scaling would be largely mechanical and would not require much skill. But many logs are cut and piled which are partly rotten, crooked,. 66 FOREST MENSURATION. or seamy. ‘They must be entirely discarded or reductions must be made for imperfections when the contents are calculated. Skill is required in deciding what logs should be thrown out. The obviously rotten logs are not piled on the skidway at all. The contractors include many which are doubtful, and which they think may be accepted by the purchaser. The final decision rests with the scalers. There are many logs having center rot or rot only on one side, seamy, shaky, and crooked logs, which contain enough good lumber to pay for the hauling, but cannot be given a scale equivalent to straight sound logs of equal di- mensions. When such a log is measured, a deduction is made to compensate for the loss through the imperfection. If the scaler is recording only the diameters and lengths of the logs, discount for defects in a specified log is usually made by reducing the measured diameter sufficiently to cover the loss. Some- times, chiefly in the South, the allowance for defect is made by reducing the log’s length. If the contents of the logs are reduced in the woods, the discount in board feet is made when the log is. measured. The experienced scaler who has worked at a saw- mill is able to estimate the loss through certain imperfections merely by inspecting the log. It requires skill and experience to recognize defects and to know how much they affect the quality of the timber. It also requires good judgment to determine how much the dimensions of a defective log should be reduced to scale what can actually be manufactured from it. The best scalers have this experience and judgment. Many, however, make deductions for defects largely by guesswork. The writer has encountered a few rules for special cases, but there is apparently no uniformity in practice among different scalers. This lack of uniformity is unfortunate, and while it is impossible to lay down rules which are universally applicable, it is possible to classify the principal problems met by scalers. It would be entirely practicable for lumbermen to follow a uniform system of handling these problems, making modifications as required in special cases. Discount for Center Rot.—If a log has a rotten spot at the center, and there is enough good wood to pay for hauling, a dis- METHODS OF SCALING LOGS 67 count for the defect is made in the scale. Several incorrect methods for computing this discount are in use. One method requires the subtraction of the diameter of the rotten core from the diameter of the log for the required diameter. Thus if a 12-foot log were 20 inches in diameter, and the rotten core had a diameter of 6 inches, this method would make the new diam- eter 14 inches. The loss (using the Champlain Rule) would be 122 board feet, which is ridiculous. Another method is to scale the log as sound, compute the contents of a log the size of the core, and subtract this from the scale of the log. In case of the 20-inch log with a 6-inch center rot the loss would be 17 board feet. Another scheme is to add 3 inches to the diameter of the rotten core, square this and deduct from the gross measure- ment. The result, if the method be applied to the example above, would show a loss of 81 feet. The actual loss, as shown by a diagram, would be 33 board feet. This shows that some of the methods of scaling in practice are thoroughly incorrect. The writer has for some time considered the possibility of “CULL TABLES” to assist scalers in making discounts for specified sorts of defects in logs. In pursuance of the idea of basing such “CULL TABLES” on diagrams, the writer secured the services of Mr. H. D. Tiemann, of the U. S. Forest Service, to experiment with the construction of the tables. In the first place Mr. Tie- mann constructed a series of diagrams representing the cross- sections of logs of different diameters and calculated the actual loss occasioned by center holes of different sizes. In construct- ing the diagrams, 4 inch was allowed for saw-kerf and 4 inches as the width of the narrowest board. It was assumed that the logs would be sawed so as to yield the greatest possible output. Experiment showed that the most is obtained by “sawing through _and through” up to a certain point where the holes are large enough to make “sawing around” necessary. It was recog- nized also that in sawing through and through there might be a difference whether the log is cut so as to have an inch board from the center or to have the saw pass exactly through the center. In every case the maximvm yield was used. 63 FOREST MENSURATION. Mr. Tiemann’s study established the fact that in logs of the same length, the loss due to holes of any specified size is practically unijorm, regardless of the diameter oj the log. ‘This law is clearly shown in the table below. It happens that for 12-foot logs this loss is almost exactly expressed by the formula "(D+1)2, where D is the diameter of the hole. LOSS BY CENTER-ROT IN TWELVE-FOOT LOGS OF SELECTED DIAMETERS AS SHOWN IN DIAGRAMS. 12-inch Log, 16-inch Log, | 24-inch Log, 36-inch Log, 48-inch Log me Loss when Loss when | Loss when Loss when Loss when iu Sawed Sawed Sawed Sawed Sawed E = Thro gh| Around |Thro gh| Around |Thro’gh| Around |Thro’gh/Around |Thro’gh|Around Ins.| Board Feet. Board Feet Board Feet. Board Feet. Board Feet. 2 6 14 6 | 12 4 12 2 16 | 3 9 14 9 12 9 12 9 26 Ee ; 28 14 13 20 II 12 12 26 II 6 re 10 69 50 76 52 45 54 47 70 51 Ween ty Wikey ieee 112 82 70 92 65 80 12 Soar oe oe) EOL bins Sell REP is ws ba Rtas “ar 186 r73 194 175 | |] |) |] | | —__ |__| —__ Note.—The double lines are drawn at the points where the loss is greater by sawing through than by sawing around. In practice, logs which have holes are apt to have more loss from hidden defects than others. Therefore it is wise to allow a further loss of 5%. This gives the very simple formula of loss in board feet due to center holes: Loss = 3(D+1)?. A table showing the loss in board feet for logs of different sizes and holes of different diameters has been constructed by ae METHODS OF SCALING LOGS. 69 this formula, first for 12-foot logs and then for 1o-, 13-, 14-, 16-, 18-, and 20-foot logs, and is given on page 71. This table is applicable to all center defects, such as holes, cup shake, rot, etc., which are four inches or more from the bark. (Fig. 3, D, E, and F.) To apply the table, measure ; : Fic. 3.—Methods for Discounting the Scale for Defects. the longest diameter of the defect, find the loss in board feet from the cull table, and deduct from the gross scale of the log. If the defect runs through the log, or if it appears only at the large end, measure the defect at the large end, otherwise at the small end. The table should be used only with short logs. Some may naturally ask how one is to determine the length of a hole if it appears at only one end. It is assumed that, if a defect appears at one end, there will be a loss to the center board 70 FOREST MENSURATION. throughout the length. If short pieces can be utilized, that is the gain of the millman, and the fact is an element of conserv- atism in the rule. The same principle holds good for the suc- ceeding cull rules. Discount jor Defects near the Edge of Logs.—Under this head may be included rot, splits due to careless felling, super- ficial shake due to fire scars, sun scald, frost, or any other defects which require the removal of a wide slab, as shown in Fig. 3, A, B, and C. Cull Table B has been constructed, by the use of diagrams, to show the loss by cutting slabs of different widths from logs of different diameters and lengths. The scaler measures the width of the slab which would obviously have to be cut off, finds in the table the loss in board feet, and deducts this from the gross scale of the log. If the defect runs through the log, following the grain, and does not extend deeper at the large than at the small end, the measurement is taken at the top. If the defect appears only at the large end, or extends relatively nearer the center than at the small end, the scaler must estimate the width of the slab, at the small end, which would have to be taken off. Cull Table C is designed to meet the case of defects on the side of a log, which the sawyer eliminates by cutting around them, rather than by taking off a wide slab. These are narrow defects running rather deep into the log, such as are indicated in Fig. 3, Gand H. The loss in sawing around such a defect was found by Mr. Tiemann to be equivalent to removing a wedge- shaped piece, a sector, fully enclosing the defect, though this principle does not exactly indicate the sawyer’s method of sawing the log. On this principle Cull Table C was constructed, by diagrams, to show the loss by cutting around sectors of different sizes from logs of different diameters and lengths. To use the» table, estimate whether the defective spot is entirely included within one-sixteenth, one-eighth, etc., as represented by a frac- tion of the circumference of the end of the log, then find the loss from the table and deduct from the gross scale. Just as in the case of cutting a wide slab, the scaler must estimate, with refer- ence to the small end, the portion of the log wasted. (Good for defects more than 4 inches from the bark.) METHODS OF SCALING LOGS. CULL TABLE A. Loss BY DEFECTS OF DIFFERENT DIAMETERS NEAR THE CENTER or Loas. Length of Logs in Feet. 71 Diameter sae iia 10 | 12 | 13 | 14 | 16 18 | 20 Inches. Board Feet. 2 5 6 6.5 7 8 9 10 3 9 II 12 I 15 16. 18 4 I4 17 18 20 23 25. 28 5 20 24 26 28 32 36 40 6 27.5 33 36 38.5 44 49. 55 7 36 43 47 50 57 65 72 8 45 54 58.5 63 72 81 go 9 56 67 74 78 89 100 112 10 67 81 87 93 107 120 133 II 80 96 104 I12 128 144 160 12 94 1 122 132 I51 169. 188 13 109 131 142 153 75 196. 218 14 125 150 162.5 175 200 225 250 15 142 L71 184 218 226 255 283 CULL ‘TABLE. S.- Loss By CUTTING SLABS FROM ONE SIDE OF TEN-FOOT LOGs. Diameter of Log in Inches. Width of Slab, 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 Inches. Board Feet. I fe) fe) I I 2 2 3 3 3 4 4 2 3 4 5 5 6 7 8 8 8 10 IO 3 7 8 9 10 Il 13 14 15 15 16 17 8 Ge |e aaa 14 15 £7 +t i 20 22 23 25 26 28 ie ee es 22 24 26 28 30 33 35 a7 39 he Site ® 9) (pany re ae 33 35 38 41 43 46 49 52 BT eS | ee 45 49 52 55 59 62 65 (is Bel Ee | ee Se eee ee 60 65 68 73 76 80 OG SER oy Sees Caney tS Ca 77 82 86 gI 96 eT ae) a) OG Ae Pee emer (ares, Peet ae 97 102 | 107 | 112 SEEMS 8 2150. fo, beatae |v 6s io! of wiv ole a ftete eiellg.s ke A ele 11g | 124 | 129 PMS Shy eos) c's ate 2 Nis Sand | Wil 0 hap ell wae OR eg at ie ines als Wied 141 | 148 pe on 8 ee ee ee ee me aes OR Pe ee en eee Fe 167 72 FOREST MENSURATION CULL TABLE B. Loss By CUTTING SLABS FROM ONE SIDE OF TWELVE-FOOT LOGs. Diameter of Log in Inches, Width of Slab, | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | Inches. Board Feet. I oO I I 2 2 3 3 4 4 5 2 4 5 6 7 7 8 9 10 II II 3 9 10 Il 13 14 15 16 18 19 20 Pgs ate Ae r7 19 21 23 2 26 28 30 31 | OP) ae 27 29 2 34 37 39 42 44 Bi ete aa etceta ee ee 39 43 46 49 2 56 59 a (AER AR | SE ROREY Gem -<.- hs! BA 55 59 63 67 71 75 i ot ei drat 5 tlle eh, sa ee rae te 7% 78 82 87 g2 Oe, LM) Cringe yy Gaeec ie) Parieengs erariemes oe) 93 98 | 104 | II10 BO Che vig SS Oreo siete lic dot albcea ats to ee ees 117 4223777286 EP a Ss Ws a yee ee) Mea Noe OUY IRE me! 259 142 | 149 Bs Dota tececattaioareiatlson tate ol x ghoaa sil Gxt 6! a'fie ene te cRRERREN 5, a.cathia Sone 170 or Sma Per) aad (cue (Fig red) Smenetrd Pearse) Oe) CIE VM CULL TABLE B. Loss By CUTTING SLABS FROM ONE SIDE OF FOURTEEN-FOOT LOGs. Diameter of Log in Inches. Width of Slab, 6 | 8 | 10 | 12 14 | 16 | 18 | 20 | 22 | 24 26 Inches. Board Feet. t I fe) I I | 2 2 a 4 4 5 6 6 2 4 5 6 7 8 9 10 II 12 1a) Fa 3 10 12 13 15 16 18 19 20 22 23 25 Pe et 19 22 24 26 28 30 32 34 36 39 Sy RS, ce ee 31 34 37 40 43 46 48 51 54 Ge Wes Fotis meee tere bom 46 50 53 57 61 65 68 72 ae ee eel Pers eS. ae 64 68 ape 78 82 87 92 Bip iioate Peal Rae tetins otc pecfant w mde 85 gt 96 | 102 || 107 fare Oi Ps SS os eee ese Slo os ee hy oa ae ee 108 | 114 | 121 | 128 | 134 TO: Hs bao s alter yest fits Sls [is a ea ae 136 | 143 | 150 | 157 BM AT hath heal rece ced coe sv auleadla taints ear pe a ape 166 | 173 | 181 a CR A POR aS Pee Rre Peacemcied eee | sk Rael gee | 197. | 207 EQ Met bbs balls ae allests 00 f Saco) ep cee Peel cs 3-3) 234 26 METHODS OF SCALING LOGS. 73: CULL TABLE: Bi; Loss BY CUTTING SLABS FROM ONE SIDE OF SIXTEEN-FOOT LOGs. Inches. —_—_ — $$ —$—$—$—— Board Feet. 13 15 i 18 20 21 23 25 26 28 22 2: 2 30 32 Kh of 39 4! 44 ‘ore 35:*| 39 | 42 ) 45 4032-5277 55] 59 | 62 Ene CSE 52 57 61 65 69 74 78 83 3 2 pees See 73 78 83 89 94 99 | 105 ean (Cae geen OF 104 |} 0G Sip | 222.1128 cd Bee (eee wes (aed (kh Fas. | Rok. DA akAG 1-153 Se Ses Pera PSeeme aye egy 8 P55 || 163- |. 171~| 279 SEB A ean Pres rere ae a 189 | 198 | 207 URES Saat (Ae re eS (Acree oe eee 220.1 257 CR ORY DCOROaa ae) (CRs Dae mre Po COn Ere ry (aes taser (Le Bd ee 1875 || bes eee | 268 see Cn Au WN ve) Soe e wy, ‘6 Y 2 ae) 8 Sea w! 06 Diameter of Log in Inches. lt 2S Ee] ee ae ey of Slab, 6 8 10 12 14 | 16 18 20 22 24 | 26 fe) I I 2 3 3 4 5 5 6 Zi 5 6 7 9 10 II 12 ac 14 15 16 Discount jor Crooks.—Usually logs are supposed to be straight, and the scaler does not make any discount for crooks when he measures the logs. When logs are piled on skidways, it is obvi- ously impossible to take crooks into consideration. Often, however, a small percentage is deducted from the total scale to allow for this imperfection. To make allowance for the loss by crooks in a specified log, the scaler sights over the surface and calculates how much the small end must be reduced to circum- scribe the square piece which really can be cut from the log. Discount jor.Wormy or Rotten Sap.—The diameter measure- ment is taken inside the sap, that is, the heart-wood alone is scaled. | Discount jor Seams and Shakes.—Seamy and shaky logs are usually culled altogether. Sometimes in a tree with straight grain, a seam causes only the loss of one plank in the center. This loss may be calculated by the rule: Multiply the thickness of the plank to be discarded by the diameter of the log, multiply by the length and divide by 12. Usually the grain of the log is not straight and it has to be discarded altogether. 14 FOREST MENSURATION. CULL TABLE C. Loss FROM DEFECTS CONTAINED IN SECTORS REPRESENTING FRACTIONS OF LOGs. Diameter of Log in Inches. Length| Part of |~ : eae bai j ot Circle | 6 | Ss | 10 12 | 14 16 18 20 22 24 26 Log, | Re- | | Feet. moved. ; = nh are hae Board Feet. 10 is | 2 3 5 6 8] 11 14 17 20 25 30 4 4 6 8 | 31 14 | 18 22 28 33 41 49 } 7 40 7 25%) 20.) ae 2 40 49 61 75 go 3 7 |-¥4 1-22] 301 301.49 |) 62 | 77 | ‘Qader t245) ome 4 7 | 14] 22 | 33 | 45 | 60 | 77 | 97 | 110) 1414 ee 12 ts 2 4 6 Sf yor 1 ae 17 20 25 30 36 $ Sle 74 70,| 13 eee ae). 33) 4a eee t 9 | 13] 18] 24] 30] 39] 48] 59| 73] 90] 108 3 9 | 17-1 27 | 36.) 47} 59) Fsu. 93] 113i) Bae 2 9 | 17.|-27 | 39.) 551 73 1, OS) 217 | 142) ee 14 ts 2 4 6 9| 12] 15 | 20 2 29 34 42 $ 6] 8} 32) 35 }99.) 25< Shy 39 | 47.) See 4 ¥O 1°15 | "20° } 27: -35>1 45 56 69 85 | 105 | 126 3 IO) IO ah Nar) 540} 66 87 | 108 | 132 | 159 | 187 4 10 | 19 | 31 | 46 | 64 | 85 | ro8 | 136 | 166 | 197-1) aan 16 16 3+ S)..7 1.20 ).13 b 7 |e 27.) 33.) aes $ 7 | 91 23 1.27 | 22-| 28°) ga -44*) 52°) 78 t RTs] -B7.4°23) 1334140 | SP | 8 79 | 197. | 3 oe ee 2 IX | 22-1) 35.) 471 6241 79 90.} 123 | 151 | rates 4 re) ax 35 2'| 731-97 | 324] 355 | 199} 226°) 20m Shaky logs are usually valueless. If the shake is confined to the center, the cull rule for center-rot may be used. 40. Rules for Scaling Used on the Forest Reserves.—The following rules * have been issued to the Federal forest officers to govern the scaling in timber sales on the forest reserves: All timber must be scaled by a forest officer before it is removed from the tract or from the points where it is agreed that scaling shall be done. Each stick of sawlogs, timbers, poles, and lag- ging must be scaled separately. Rough averaging of diameters or * From The Use of the National Forest Reserves, U. 5S. Dept. of Agri- culture, Washington, D. C., 1905. Oe METHODS OF SCALING LOGS. 75 lengths is not allowed. The Scribner rules will be used in all cases. Ties may be actually scaled, or reckoned as follows: Kight-foot ties, standard face, 334 feet B. M., each; 6-foot ties, standard face, 25 feet B. M., each. Shake and shingle-bolt material is measured by the cord. Squared timbers are scaled by their actual contents in board feet with no allowance for saw-kerf. Thus, an 8 12-inch 16-foot stick contains 128 feet B. M. Unsound or crooked logs will be scaled down to represent the actual contents of merchantable material. All partially unsound but merchantable stuff must be scaled, whether removed or not.: In ground-rotten timber, butts which, though unsound at heart, contain good lumber toward the outside, are frequently left in the woods. Where such material will pay for sawing, the forest officer will scale it at what he considers its true value and include it in the amount purchased. Logs which are not round will be scaled on the average diam- eter; flats and lagging on the widest diameter. In the absence of a log rule, or where the position of logs in the pile makes its use difficult, the diameters and lengths may be tallied and the contents figured from a scale table later. When possible, the purchaser will be required to mark top ends of logs to avoid question when they are scaled in the pile. The forest officer should insist on having one end of piles or skidways even, so that ends of logs may be easily reached. When the lengths of piled logs are hard to get, two men should work together. When scaled, each stick of sawlogs, timbers, ties, lagging, posts, poles, or piles must be stamped with the United States mark on at least one end, and on both when possible. Cord material, such as wood or bolts, must be stamped at both top and bottom of piles, and at least 12 pieces in each cord must be stamped. All scaling is inside of bark. CHAPTER VI. DETERMINATION OF THE CONTENTS OF LOGS IN CUBIC FEET. 41. Use of the Cubic Foot in America.—The cubic foot is already used extensively in forestry in the United States. Many of the most useful tables of contents of standing trees, of growth, and of yield, have been obtained by the use of the cubic foot. Although most figures of volume are finally expressed in board feet or other unit common in commerce to have practical value, the cubic foot is often the basis for these results. Board measure, cord measure, and standard measure are useful only in buying and selling wood and timber. ‘These units can never be used satisfactorily in scientific work where the exact contents of logs and trees are required. The uses of the cubic foot in preparing: volume tables for standing trees in cords, board feet, and standards, and in determining the laws of growth of trees under different circumstances are explained in later chapters of the book. The cubic foot is but seldom used in this country for buying and selling round logs. It is, however, used with high-priced imported woods, occasionally with hickory and oak, and with squared timber. The cubic foot will unquestionably be used more and more, as the value of timber increases, and eventually in large measure replace the present rough unit, the board foot. The American forester must, therefore, be familiar with the principles and methods of determining the cubic contents of logs and trees. 76 CONTENTS OF LOGS IN CUBIC FEET. 77 42. The Measurement of Logs to Determine their Cubic Con- tents.—All the methods of cubing logs require the measurement of the diameter at one or more points and the measurement of the length. Ordinarily the diameter measurements are taken with calipers. Formerly in India and in Europe the circumference was measured with a tape. In this country a tape is only used when calipers cannot be obtained, and in work with the trees of the Pacific Coast which are too large for ordinary calipers. A rule is sometimes used where ends of logs are measured inside the bark, as when a study of growth is being made. Generally the measurements for volume are taken in the fol- lowing way: Two measurements are taken with the calipers, one a Fic. 4. b giving the greatest and one the smallest diameter. (Fig. 4a.) The average is considered the average diameter of the log at the point measured. If the log appears to be perfectly round, only one measurement is taken. Some attempt to measure one average diameter, even when the log is not round. (Fig. 4b.) Sometimes the log is in such a position that it is difficult to measure accu- rately the longest and shortest diameters, as when it is lying on a flat side or is sunken in a depression. In such cases the log must be moved or rolled over, if an accurate measurement is to be taken. If the measurement of diameter inside the bark is sought, the bark may be chipped off or its average width may be determined by 78 FOREST MENSURATION. separate measurements and deducted from the diameter outside the bark. Sometimes the diameter outside the bark is desired at a point where the bark is torn away altogether or in part. Then the diam- eter inside the bark is measured and the average width of bark added, the latter being determined from a neighboring part of the log. Where volume measurements alone are sought, the common practice is to measure the diameter outside the bark, and to de- termine the inside measurements by deducting the bark width. This method is not so accurate as measuring first inside the bark and adding the bark width. The reason for this is that there are apt to be irregularities of bark which make the measure- ments of diameter too large, or pieces are broken off and the diameter is too small. This is particularly true with trees with soft scaly bark like the yellow pines. With hardwoods the common method of measuring outside the bark is accurate enough for most purposes. In determining the average thickness of bark, several measure- ments should always be taken at different sides of the log, unless the bark is thin and obviously uniform in thickness on the entire circumference. Where the bark is deeply cut, like that of an old pine, each measurement of width should show the thickness between the wood and a line tangent to the log, as the arm of a caliper fitted to the log at that point. : Sometimes there is a swelling due to a knot or other cause at the point where it is desired to take the measurement of diam- eter. In this case the calipers must be placed just above or below the swelling, or measurements may be taken both just above and below and the average called the correct diameter. Many inaccuracies arise from the careless use of calipers. They should always be placed at right angles to the log. Inas- much as the sliding arm of the calipers when moved bends toward the stationary arm, as explained on page 81, care must be taken that it rest tightly against the log and is at right angles to the graduated beam when the reading is taken. Whenever possible, the calipers should be placed so that the beam touches the log. CONTENTS OF LOGS IN CUBIC FEET. 79 The measurement should not be taken with the tips of the arms, except in extreme cases, because the measurer is apt not to have the sliding arm brought to a perpendicular position with the beam, and because any inaccuracies of the calipers due to warping or wear are greater at the tip than near the base of the arms. The forester should continually test his calipers to guarantee their accuracy. In all scientific work diameter measurements are taken in inches and tenths. Foresters are sometimes tempted to round the measurements to half or to whole inches, especially when a large number of logs are being measured. If the cubic foot were used in commerce, the diameter measurements would probably be rounded to the half or whole inch, as is the case where logs are scaled in board measure. At present in this country the measure- ments of cubic volume are chiefly for scientific purposes, as in the preparation of volume tables, the study of growth, etc., and require diameter measurements to tenths of inches. It is frequently asserted that there are apt to be inaccuracies due to irregularities of the bark, and that in consequence it is incon- sistent to take such fine measurements. It is true that there are chances for errors in measuring logs with rough or ragged bark, but this is no reason for deliberately adding to the errors by a rough method of taking the readings from the calipers. Moreover, the rounding of diameter measurements to half or whole inches leads invariably to carelessness on the part of the measurer, who may soon estimate certain diameters without using his calipers, or in other measurements give figures which are estimates rather than true readings from the instruments. 43. Measuring Instruments.—In work on very large logs like redwood logs, circumferences are taken with tapes. A special tape is made for such work, which shows not only the cir- cumference in inches or in feet, but also the diameter correspond- ing to every circumference. The readings, therefore, may be recorded as diameters, thus avoiding the laborious work of after- wards calculating the diameters from circumference readings. So FOREST MENSURATION. The end of the tapé is provided with a pin which may be inserted in the bark, enabling one person without assistance to measure a large log or tree. (Fig. 5.) Usually the tape gives a larger result than calipers, because every swelling or abnormal protuberance of bark is included in the tape measurement. It is impossible to bring the tape close against the trunk at all points. Care should, therefore, be taken in using the tape to avoid irregularities on the log which may affect the measurements. Where the diameters of the ends of the logs are measured, a rule may be used. This is often done in taking full tree Fic. 5.—Tape for Measuring Girths and Diameters. analyses. \'The diameters inside the bark are measured with the rule on the smooth cross-cut, and the outside dimensions obtained by adding the bark width. For these measurements the cross- cut must be made at right angles to the axis of the log, otherwise the figures will be too large. A number of different forms of calipers are made. American foresters generally use the type of calipers developed by the Forest Service, U. S. Department of Agriculture. This form is extremely simple, and has been found to be light, strong, and durable, as well as very accurate, the qualifications necessary for a satisfactory instrument. These calipers consist of a beam CONTENTS OF LOGS IN CUBIC FEET. 81 having scales on both sides graduated in inches and tenths. This beam is provided at one end with an arm held in place by a bolt and nut, which permit it to be detached for convenience of transportation. ‘The beam is provided with a sliding arm fitted loosely so as to slide easily over it, but constructed so that when pressure is applied to its inner edge, as when it is brought against v tree-trunk, it swings into position in which it is at a right angle to the beam. For use in eastern forests the most convenient caliper has a beam measuring 36 inches and arms one half that length. In forests where trees over 3 feet in diameter occur, AC WLP A iH HU )UCUUFUSAH LY ULES UL PL = 2} 3) 4 St él al 8) 9] : 7 mm MMMM I MAT MTN Te IME INTIMAL 13 14119 116117118 /119'2)911'29'913'24'95 126197918 919'30'31 32/33 3431036 Fic. 6.—American Type of Calipers. calipers having a beam measuring 50 inches and proportionately long arms are used. The 36-inch calipers weigh 1.9 pounds. The upper and lower edges of the opening in the sliding arm are lined with metal to prevent wear. The metal strip lining the upper edge is movable at one end, being held in place by a screw (A in Fig. 7). This device enables the perfect adjust- ment of the arm with reference to the beam of the calipers. The chief disadvantage of the American type is that the space for the beam in the sliding arm is made so narrow that the $2 FOREST MENSURATION. latter does not run smoothly or it actually sticks when the beam swells, as often occurs if used in the rain. The same thing hap- pens if the beam becomes slightly coated with pitch, which cannot be avoided when working with pine logs; and again the sliding arm is apt to be clogged by damp snow and seriously interfere with winter work. This disadvantage is obviated in the calipers used in Germany, described below. The German calipers are, however, heavier and for most work in this country less convenient than the American type. This form of calipers is manufactured by Keuffel & Esser Co., No. 127 Fulton Street, New York City, listed at $4.50 each. The calipers used in ordinary forest work in Austria are very similar to the American type just described. The device for adjusting the sliding arm differs only-in the position of the retain ing-screw. The graduations on the beam are marked in aepaes sions, about an inch wide, in order to protect the marks from the wear of the sliding arm. As ordinarily constructed, the calipers are heavier than those made in this country and have the disadvantage of not taking down. A number of different kinds of calipers are used in practice in Germany. Simple calipers like the Austrian type above described are used by many foresters. A common form is that made by Staudinger & Co., in Giessen, which has a special con- CONTENTS OF LOGS IN CUBIC FEET. 83 struction designed to obviate the difficulty caused by the swelling and consequent sticking of the movable arm. In these calipers the measuring-beam is beveled on the edges so that the cross- section is a regular trapezium. Fig. 8 shows a section of the movable arm AA, including a cross-section of the measuring-beam M. The construction is such that at the points a, a’, and a” the measuring-beam fits closely to the sliding arm, but does not come in contact with it at any other points. The sliding arm is further fitted with a metal wedge shown in cross-section as N. This wedge is held in place by the screw 0, and whenever the screw turns is moved toward or away from the measuring-beam. The screw 0 is turned by means of a key, which is provided with two points made to fit the shallow holes in the head of the screw. By loosening or tightening the wedge the sliding arm may be adjusted to suit the condition of the beam. If it is swollen by moisture or coated with pitch, the wedge may be loosened so that the arm will move with ease. The disadvantage of the calipers is that there is a separate 84 FOREST MENSURATION. key, which is easily lost. The construction being less simple than in the forms described above, the weight is necessarily greater. Fig. g shows another adjusting arrangement, some- times found on the sliding arm of German calipers. A number of folding calipers are constructed for convenience in packing and in carrying to and from work. The author has never seen any folding calipers which were serviceable for prac- tical woods work. ‘They are not so strong as the regular forms, and with use they soon become inaccurate. A caliper which may be taken down like that first described has every advantage of the folding caliper except that it cannot be readily taken apart for transportation to and from daily work. This last advantage is too insignificant to require any sacrifice in the strength and durability of the instrument. Every European text-book on Forest Mensuration contains descriptions of calipers constructed on other principles, as, for instance, like a carpenter’s caliper-gauge. ‘The author has never found any such calipers in extensive practical use even in Europe. : The length of logs is usually taken with a tape graduated in feet and tenths. The length is taken along the surface of the log. If the log is considered a frustum of a cone or paraboloid, this represents the slant height and not the true length of the axis. The error is, however, very minute and may be disregarded. On an average this amounts to only 0.1%, as has been proven by European experiment. It was explained on page 63 that in scaling logs for board feet the length is usually rounded to feet and always to the foot below the actual measure, as, for example, 16 feet, if the actual measure is 16.7 feet. In all scientific work with the cubic foot, the measurements are rounded to tenths of feet or to inches, preferably the former. For general work of forest measurements a steel tape, measur- ing 50 feet, is the most satisfactory. It is convenient in size and weight, and is more durable than any other form of tape. A CONTENTS OF LOGS IN CUBIC FEET. 85 metallic tape, that is, the cloth tape which has several strands of copper wire running through it, is satisfactory in every respect, except that it frays after a short time when used in the woods. A steel tape will last several years in constant use, provided it is not allowed to rust. After using a steel tape in wet woods, it should always be wiped and oiled. A number of different forms of steel tape are constructed. The best type has a band ? inch wide, and costs $6.50. A cheaper tape with a band } inch wide costing $4.45 answers every purpose, but will not stand as rough wear as the more expensive form. With careful treatment the smaller tape should satisfy the requirements of most foresters. Metallic tapes cost $2.75. Cloth tapes are impractical for ordinary rough work in the woods. Tapes may be purchased from any dealer in survey- ing instruments. 44. Principles Underlying the Determination of the Cubic Contents of Logs‘and Trees.—For many years European foresters have endeavored to discover a mathematical formula by which the cubic contents of logs may accurately be calculated from a few measurements. Great difficulty has been encountered, because the forms of different logs differ so much under different con- ditions. The form of a specified log depends on its relative growth in diameter at different points. The growth in diameter at different parts of logs varies widely, and in consequence their forms are not uniform. If the diameter growth on the trunk at certain distances above the ground were always the same for a given species, the form of the trunks of all trees of that species would be constant. But the form of a tree changes from decade to decade, different individuals nearly always differ in form, and logs from different parts of the same tree have different forms. At first sight the surface lines of a log appear to be perfectly straight, as on the section of a cone. Experiment has shown that on most logs the longitudinal surface lines are slightly convex, as on the section of a paraboloid. Sometimes, however, they are slightly concave, in which case 56 FOREST MENSURATION, the log approaches the form of a Neilian paraboloid or neiloid., Usually the form of the log is between that of a cone and a paraboloid. A number of formule have been devised which enable the cubing of logs with almost perfect accuracy, the error amounting to less than one percent. The most accurate formule, however, require for their use too many different measurements on the logs, or they involve too many calculations, to be of use in practical work. In ordinary work in the woods extremely simple formule are used, which are accurate enough for commercial purposes, although they are subject to an error in individual cases of 2 to 4 per cent. 45. Fundamental Formulze.—It is customary to assume that logs and other parts of felled trees have the form of some known geometric body, as the frustrum of a cone or paraboloid, and to cube them by formule applying to these bodies. The methods will be clearer if prefaced by a“statement of the most impor- tant formule for cubing a cylinder, paraboloid, ce and neiloid. ‘These formule are as follows: FORMULAE FOR DETERMINING THE VOLUME OF A CYLINDER, CONE, PARABOLOID, AND NEILOID. The Cylinder. | Let V=volume of the cylinder; Hf =altitude of the cylinder; R=radius of the base; D=diameter of the base; B=area of the base; 1. V=2zkR?-H; 2 >: Vie 4 CONTENTS OF LOGS IN CUBIC FEET. The Full Cone. Let V=volume of the cone; H =altitude of the cone; R=radius of the base; D=diameter of the base; B=area of the base; D,= diameter at 4H; By =area of cross-section at 47; By=area of cross-section at 4H; \ ' ' wm be ee we oe ee ee eww ~b- eg ! ' ! / zR?-H nh ii = “ = 3 FAG. rr. , Ser, be ak Ke ow), _ ‘ N 7 ‘ 1% \ \ Ne NW 4 \ & “Ry “Shee ty, Fic. 20. distance CD, which, subtracted from the value BD, gives the height of the tree. Another case is where the observer stands on a slope and measures a tree below. (Fig. 20.) After determining AD, he sights the instrument to C and finds the value of CD, namely, the distance from the level of the eye to the base. Then he sights to B and obtains the distance between the tree’s tip and the level of his eye. This reading subtracted from the first gives the total height of the tree. In using the instrument, a point of observation is chosen where the tip and base of the tree can be distinctly seen; then the 128 FOREST MENSURATION,. horizontal distance to the trunk is accurately measured with a tape. The slide of the instrument is set so that the distance between the point of suspension of the plumb-line and the hori- zontal scale corresponds to the distance from the observer to the tree. If this last distance is less than 20 units (feet, yards, meters, or other unit), the slide is set with the index mark IT pointing down, and the right-hand vertical scale is used. If the distance from the tree is more than 20 units the slide is inverted, and the left-hand vertical scale is used. The observer then takes the instrument in the left hand and sights to the tip of the tree, the right hand holding the folding mirror and at the same time steady- ing the instrument. When the plumb-line comes to rest the point of intersection with the left-hand horizontal scale is read in the mirror. This reading is the distance from the level of the eye to the tip of the tree. If the base of the tree is below the level of the eye, the observer sights the instrument to the foot of the tree and takes a reading from the right-hand horizontal scale, which gives the distance below the level of the eye. This must be added to the first reading for the total height of the tree. If the base of the tree is above the level of the eye, the observation is first taken to the tip of the tree and then to the base and the latter subtracted from the first. If the tip of the tree is below he level of the eye, an observation is taken to the base and then to the tip, the difference being the total height. | The Faustmann height measure is compact, light, and well adapted for rough work. The only delicate part is the mirror, which folds against the face of the instrument; it is not apt to. be broken except by very careless usage. It is extremely accurate when used by a trained hand. With practice one should be able to measure trees not over a hundred feet high within a foot and those not over 50 feet high within 6 inches. Trees above 100 feet can be measured accurately, provided one can find a point of observation where both the tip and the base can be easily seen. When the instrument is in constant use, it is necessary / DETERMINATION OF THE HEIGHT OF STANDING TREES, 129 to renew the thread frequently, as it is apt to become frayed and cause inaccurate readings. It is difficult to use the instrument in a strong wind, because the plumb-line is light and easily moved. But, as most work with a height measure is in the woods sheltered from high winds, this is not a serious objection to the instrument. The steel instrument costs $19.50 in this country and 35 marks in Germany. ‘The wooden form costs in Germany from 6.50 to ‘ i iN Fic. 21.—The Weise Height Measure. 12 marks and $6.50 in America. They may be purchased from Keuffel & Esser Co., N. Y., W. Spodrhase, Giessen, and L. Tesdorpf, Stuttgart, Germany. 68. The Weise Height Measure.—This instrument consists of a metal telescope barrel g inches long, fitted at one end with a 130 FOREST MENSURATION, peep-sight and at the other end with stout cross-wires. The former is on a separate tube which fits closely in the telescope barrel. A strip of metal is fixed tangent at its right-hand edge to the periphery of the barrel. On this strip is graduated a right- hand and left-hand scale, which meet at a zero point near the objective. The right-hand scale has 47 and the left-hand scale ro graduations. ‘There is a notch at each graduation mark indenting the outside edge of the strip. A detachable metal bar, square in cross-section, fits in an opening in the plate at the zero point of the scales, and is held in place by a retaining-spring. A pendulum whose rod is triangular in cross-section is suspended by a universal joint from the upper end of the bar. The sliding bar has on one side a scale with graduations corresponding to those of the metal strip. When the instrument is not in use, the sliding bar is detached and packed inside the barrel. This height measure is constru#@ted on the same principle as Faustmann’s. The scale on the strip corresponds to the horizontal scale of the Faustmann and the graduated bar corresponds to the vertical slide. The instrument is used in the same way as Faustmann’s. After selecting a station whence the tip and the base of the tree may be seen, and measuring the horizontal dis- tance from the tree, the sliding bar is set to correspond to the distance from the tree. The observer holds the instrument in his right hand, and, as he sights to the tip of the tree, turns it over to the left enough to allow the pendulum to swing free from the notched strip. As soon as a satisfactory sight is obtained, the instrument is turned over to the right and the pendulum caught in the opposite notch. The reading taken at that point shows the distance from the level of the eye to the tip of the tree. The distance of the foot of the tree below the level of the eye is then obtained and added to the first result as the height of the tree. The Weise height measure is very compact and strong and therefore well adapted to forest work. The chief advantage of . the instrument is that the notches catch the pendulum and hold | it in place when the sighting is completed, thus obviating the = ses THE HEIGHT OF STANDING TREES. I3z disadvantage of a limber line which may alter its position after the eye leaves the sight. When used by a practiced hand it is very rapid, probably somewhat more so than the Faust- mann height measure. On the other hand, the notches prevent the possibility of a reading closer than 4 unit. It is therefore not as accu- rate as the Faustmann height measure. The instrument may be purchased from W. Sporhase, Giessen, Germany, price (in Ger- many) r2‘marks; L. Tesdorpf, Stuttgart, Ger- many, price (in Germany) 13-15.50 marks. 69. The Christen Height Measure.—This instrument consists of a metal strip 16 inches long, of the shape shown in Fig. 22. The in- strument shown in Fig. 22 is made of two pieces hinged together, which are folded when it is not in use. A hole is pierced in the upper end, from which it is suspended between the fingers. Along the inner edge is a scale which gives directly the readings for heights. The instrument is used as follows: A 10-foot pole is set against the tree. The observer stands at a convenient station whence he can see the tip and base of the tree and also the top of the 1o-foot pole. The instrument is suspended before the eye and moved back and forth until the edge 6 is in line of vision to the tip of the tree and the edge c in line of vision with the base. The point where the line of vision from the eye to the top of the 1o-foot pole intersects the inner edge of the instrument indicates on the scale the height of the tree. Each instrument is constructed for use with a. Fi; 22.—The Christen Height specified length of pole. The instrument de- Measure, 132 FOREST MENSURATION. scribed above is one designed by the author for convenience with the use of English units. It was constructed in the following way: The distance bc on the instrument was chosen arbitrarily as 15 inches and the length of the pole as 10 i iN \\Y FIG. 23. feet. It would, of course, be possible to construct an instrument for a pole 12 feet or any other length and on a basis of any desired length of instrument. ‘The theory of the con- struction of Christen’s instrument may be shown by Fig. 23. DETERMINATION OF THE HEIGHT OF STANDING TREES. 133 When used as above described, two pairs of similar triangles are b formed: ABC, and Abc; ADC, and Adc, in which BC = Xe and do = 7. With a known value of DC and bc, dc may be determined for all different heights which are likely to be required. Thus it may be assumed that it would not be necessary to measure trees less than 20 feet high, so that the lowest gradua- tion on the instrument is made for that height. To find the proper point for the 20-foot graduation on the scale, the following formula was used: BC be 20, BS 150 . == or —=— or d¢=——=5.7 inches. 3 OG o dc 20 This same method was used to determine the value of dc\for @ 25-, 30-, 35-, 40-foot tree, etc., up to 150 feet, and the proper graduations made on the scale. The scale is somewhat more easily read when a notch is made at each graduation. The instrument is light and compact, and with practice can be used very rapidly, provided one has an assistant to manage the to-foot pole. It requires no measurement of distance from the tree and the height is obtained by one observation, whereas in the instruments already described two measurements are necessary except when the base of the tree happens to be exactly on a level with the eye. It is more rapid than either the Faust- mann or Weise instrument. Its disadvantages are that it requires a very steady and prac- ticed hand to secure accuracy; that it cannot be used accurately for tall trees; and that it is not adapted for steady work because it is extremely tiresome to hold the arm in the position required. This last objection may be overcome by using a staff to support the hand. 70. The Klaussner Height Measure.—The base of this instru- ment is a flat metal rule 6 inches long, at one end of which is 134 FOREST MENSURATION. hinged a sighting-rule slightly longer and thinner than the base and with one side cut out and beveled to a sharp edge. [ach of these two rules has a hair-line sight at its further end. At their joint is a revoluble peep-sight, which can be directed by a milled disc to either of the two hair-lines. ‘The sighting-rule may be raised by means of a high-pitch thumb-screw attached to the base-rule near the joint. The base-rule is graduated into 50 Fic. 24.—The Klaussner Height Measure. equal parts, each divided into halves, and forms the distance scale; the zero-point being at the joint of the two rules. Attached to the base-rule is a closely fitting slide carrying a thin metal strip, which is always kept in a vertical position by a weight. This strip is graduated like the base-rule, and constitutes the alti- tude scale. The instrument has a jointed ferule with clamp screw threaded to fit an ordinary camera tripod. In use, the instrument is set on a tripod at a station whence the tip and base of the tree can be distinctly seen. The oblique DETERMINATION OF THE HEIGHT OF STANDING TREES. 135 distance from the eye to the foot of the tree is measured and the slide is set at a point to correspond to this distance. The base rule is sighted to the foot of the tree and the sighting-rule is then raised by turning the thumb-screw until the tip of the tree may be sighted. The height of the tree is read from the altitude scale at the point where it is intersected by the sighting-rule. =o _ —— - -- -- -- -- ~~ ae ~~ -- ~ OO ae —— - -— -—— -- -—— — — _~ ar 4 ae ee z coertem Viena ee eae ad Nels POO aA ese we NE ge LEO net Nota ee Whe 1p Nea enw nad pnt rate ~« vat Jaca ~-— ms The theory of the Klaussner height measure is clear from Fig. 25. The instrument is set so that Ac represents the number of feet, yards, or other units in the distance AC. The triangle, formed by the base- and sighting-rules and the altitude scale, is similar to ABC, and Se Ac has the same number of units as AC, so that the number of units in bc is the height of the tree. The chief advantage of the instrument is that it is fitted to a tripod and is therefore not subject to the error due to the shaking of the hand or to an unsteady eye. It is, therefore, the most accurate of all the small instruments. A second advantage is 136 FOREST MENSURATION. that only one observation is required. It is not so compact as the instruments already described and it is more easily thrown out of adjustment. It is particularly well suited to work of a scientific character which requires accuracy. In most rough Fic. 26.—The Winkler Height Measure. forest work, hand-instruments are preferred on account of the burden of transporting a tripod. The Klaussner height measure may be purchased from Keuffel & Esser Co., N. Y., for $26.00, or from Wm. Spoérhase, Giessen, Germany for 40 marks. DETERMINATION OF THE HEIGHT OF STANDING TREES. 137 71. The Winkler Height Measure and Dendrometer.—This instrument consists of a shallow box 54 inches long, 3 inches wide, and 1 inch deep. Against one face of the box is attached a metal plate on which are marked off vertical and horizontal lines, making a series of squares. The horizontal lines are further subdivided and constitute a series of altitude scales. Each horizontal line has a right-hand and left-hand scale meeting at a zero-point to the left of the center of the instrument. The vertical line passing through the zero-point of the altitude scales is graduated as a distance scale. Suspended from the zero- point of this distance scale is a narrow flat metal pendulum bevelled on one side to a sharp edge and carrying at the end a short line and plummet. A scale is graduated on the pendulum in the same units as the altitude scales. On the upper edge of the instrument are two upright plates, the one having a peep- sight and the other fitted with cross-hairs, constituting together a line of sight. The instrument is mounted on a jointed stand- ard, which may be fitted on a tripod or held in the hand. By the construction of the instrument the observer must stand at 20, 40, 60, 80, or 100 feet from the tree. There is no sliding distance-scale, as in the Faustmann height measure, but the different horizontal lines correspond to specified distances from the tree. If the observer stands 100 feet away from the tree the too-foot altitude scale is used. If the observer is 80 feet from the tree, the 80-foot altitude scale, that is, the second one from the bottom, is used. When the instrument is sighted to the tip of the tree the intersection of the right-hand scale with the metal strip marks the distance from the level of the eye to the tip. The distance from the level of the eye to the base of the tree is obtained from the left-hand scale by sighting to the foot of the tree. If this instrument did not have the box construction necessary for its use as a dendrometer, it would be one of the most practical of all height measures. As it is ordinarily constructed, the graduations of the altitude scales are made for every five units, giving readings which are too rough for precise work. This 138 FOREST MENSURATION. objection could be easily obviated by a finer division of the scales. Another objection to the instrument is that the observer has a choice of only a few specified distances at which he must stand. This could be obviated by adding other horizontal lines, one for each 10 feet instead of 20 feet. MT meat | Fic. 27.—The Winkler Instrument used as a Dendrometer. When the Winkler instrument is used as a dendrometer, a special line of sight is used. The ocular consists of a minute hole in the metal plate which covers the right end of the instrument. At the other end of the instrument and inside the box is an objective opening in which are fitted, per- pendicular to the axis of the instrument, two metal plates, M and N (Fig. 27), of which N is stationary and M may be moved by means of the thumb-screw B. Attached to the plate M is a DETERMINATION OF THE HEIGHT OF STANDING TREES. 139 yernier plate which moves over a diameter scale marked on the face of the instrument and indicates the distance between the two plates M and N. When used‘as a dendrometer the instrument is placed on a tripod in a horizontal position, as shown in Fig. 27. A station is chosen at 20, 40, 60, 80, or 100 feet from the tree, where the point on a tree whose diameter is required can be distinctly seen. ‘The observer then tips the instrument and sights through the box to the point to be measured. The thumb-screw is turned on tetra ‘ yeh dine ees yean? ’ Fic. 28. until the two objective plates exactly enclose the trunk of the tree at the required point, and the distance between the metal plates is read on the diameter scale. The instrument is then turned over and used as a height measure. The oblique dis- tance from the eye to the observed point on the tree is determined by means of the scale on the pendulum. The desired diameter is then obtained by the formula: diameter equals oblique dis- 140 FOREST MENSURATION. tance from eye to trunk multiplied by the distance between M and N on the instrument, divided by to. lv : , . . ° Eggo , where D is the desired diameter, ¢ is the oblique distance from the eye to the tree where D is measured, and v is the distance between the objective plates M and N. The theory of the instrument is shown by reference to Fig. 28. PO is the desired diameter, pg the measured interval between the plates M and N on the instrument, and OP is the oblique distance from the eye to the tree. Two similar triangles are OP formed in which PO Op Pq. The instrument is so-constructed that the distance between the peep-sight and the objective plates M and WN is exactly 5 inches. Twenty units of the diameter scale measure § inch; each part being, therefore, ~§, of an inch. Substituting in the above formula, ¢ for OP, (uX7$y) for pq, and 35; for Op: p.V.5 120. pv Diameter = —j— = a : 1y 10 Unfortunately the instrument which is in the market is made in Austria and based on the Austrian foot and inches. The Austrian foot equals 1.04 English feet, the inches having the same ratio. In measuring the distance from the tree, therefore, it is necessary to use the following distances to conform to the Austrian measure. Austrian. _ English. a6 feet. ooo. See ees 20.8 feet oF. od RC Cate See Ay.605 Oo: fF ied OU ee 62.49 BH ates pic CARES tee 83.2 ** DETERMINATION OF THE HEIGHT OF STANDING TREES. 141 The results obtained are in Austrian inches, which may be translated into English inches by multiplying by 1.04. With proper handling the Winkler instrument gives accurate results. The degree of accuracy falls off with the increase in the height above the ground of the point measured. There is danger of inaccuracy in dark woods where the edges of the tree are not sharply defined. The instrument is much better for general woods work than any other on the market because of its compactness and simplicity. 72. The Brandis Height Measure.—This instrument consists of a square tube about five and one-half inches long, with an Fic. 29.—The Brandis Height Measure. ocular slit at one end, and a single cross-wire at the other end as an objective. To the left-hand side of this tube is attached a weighted wheel about two and one-half inches in diameter, swinging between two pivots and enclosed in a circular metal 142 FOREST MENSURATION, case. A small opening is cut in the periphery of the case, and directly opposite this opening a small lens is attached at the ocular end of the tube just to the left of the sighting-slit. The rim of the wheel, which can be seen through the opening in the case, is graduated in degrees, with plus and minus scales meeting at a zero-point, which, when the instrument is horizontal, is exactly opposite the slit. When the instrument is pointed upward or downward, the wheel remains stationary and the angles may be easily read through the lens attached to the eyepiece. On mf ¢ ‘\ a \ Pre \uU - \ ee ' | ane Ate ae Ah CREE Cee 5 oe rE mt. | Ne ! ] ee jl 7 / a Fos oe = “ay, ee Ai A Ae ee vt = ot PAGE Here 1 ne PANE scan Dect, Fide dlon RQ HN, Le st nar wygicl So FIG, 30. the side of the metal case is a retaining-spring, which clamps the wheel in any desired position, and which may be released by the ~ pressure of a small button. The theory of the instrument is illustrated in Fig. 30. AB represents the line of vision from the eye to the top of the tree; U the upper angle obtained by sighting to the tree top; AC the line DETERMINATION OF THE HEIGHT OF STANDING TREES. 143 of vision to the base of the tree; and / the lower angle obtained by sighting to the base. Then BC:AC=sin (uw+/) : sin b . BC Xsin b= AC Xsin (u +/) but sin b=cos u; substituting, BC Xcos u=AC Xsin (u+/) or _ ACXsin (u +/) % cos u ; BC For convenience a table has been constructed which gives sin (#+/) the value of the expression for all values of u and / which are likely to be required. This table accompanies the instrument. Most instruments have on the face of the metal case enclosing the wheel a small table giving the value of 20 cos a, 20 sin a, and tan a for different angles up to 30 degrees. This table enables one to compute heights when the distance from the tree is 20 meters, yards, or other units. A very complete table is published in Calcutta, entitled Tables for Use with Brandis’ Hyposometer, by F. B. Manson and H. H. Haines. Price in India, 8 annas. To use the instrument, the distance from the tree is first meas- ured, then the tip of the tree sighted through the instrument and the angle read from the wheel. Then the base of the tree is sighted and the corresponding reading taken. The value in the table corresponding to the upper and lower angles is then multi- plied by the distance from the tree, for the desired height. The height of the tree may also be obtained by reducing the degrees to tangents and multiplying by the distance, as explained in the next section. The Brandis height measure as an admir- able clinometer for measuring slopes. 144 FOREST MENSURATION. The chief value of the instrument lies in its compactness. The objections to it are that it does not give direct readings of height, but requires special computation and the use of a separate table; and that in dark woods it is difficult to read the gradua- tions on the metal rim of the wheel. It may be purchased from Max Wolz, Bonn, Germany, price (in Germany), 22 marks. 73. Clinometer for Measuring Heights.—This instrument, shown in Fig. 31a, consists of a square panel of wood recessed to receive a metal disk and a glass which protects it. The disk has | (i Fic. 31.—Goulier’s Clinometer. a curved right-hand scale and a curved left-hand scale engraved upon it below its center. These scales meet at a zero-point, and correspond to each other in their graduations, which run outward in opposite directions from the zero-point to too. The graduations of these scales represent percentages of angles instead of degrees of angles, as do the graduations of most clinometers. These two scales are swept by a pendulum ball, the lower half of which is beveled and brought to an edge having a central index mark. The upper end of the pendulum rod is formed into an eye through which a movable screw-stud passes, continu- DETERMINATION OF THE HEIGHT OF STANDING TREES, 145 ing through the disk and panel and terminating at its rear end in a push-button. A spring secured to the back of the panel engages with the button and draws the head of the screw against the eye of the rod, thus holding the pendulum fixed. When the button is pushed inward, the pendulum is free to swing by gravity when the instrument is held in a vertical plane. The instrument is only about eight inches square and may be easily carried in one’s pocket. To use the instrument the observer sights along its upper edge to the top of the tree and releases the pendulum by pressing the push-button. When the pendulum comes to rest over the right-hand scale, the pressure on the push-button is removed, permitting the spring to hold the pendulum until the reading can be taken. The number now opposite the index mark is the percentage of the angle formed by a line running from the ob- server’s eye to the top of the tree and a horizontal line running from him to its trunk. This percentage is the ratio between the height of the tree above the level of the observer’s eye and the horizontal distance from the observer to the tree. This value is multiplied by the horizontal distance from the observer to the tree, and the result is the height of the tree above the level of the observer’s eye. The observer then sights the instrument to the base of the tree, operates it as before, takes the reading from the left-hand scale, multiplies the value thus secured by the horizontal distance from him to the tree, and adds this result to that previously obtained, for the total height of the tree. These computations may be greatly simplified by taking all observa- tions at a distance of zoo feet or 100 yards from the tree. A more elaborate form of the instrument is furnished with a hinged cover having a mirror on its inner face, and with two sights located at the upper corners of the panel. (Fig. 310.) The instrument is sometimes called Goulier’s clinometer. 74. The Abney Hand Level and Clinometer.—This instrument (shown in Fig. 32) is a telescoping surveyor’s hand-level of ordinary construction, except that its spirit-tube is located above 146 FOREST MENSURATION. instead of in its main tube, which, however, contains the usual inclined steel mirror and sighting cross-wire. FiG, 32.—The Abney Hand Level and Clinometer. Combined with the hand-level is a clinometer comprising a plate screwed to one side of the main tube of the hand-level and having engraved upon it a curved right-hand scale and a curved left-hand scale. These scales are struck from the same center and meet at a zero-point, from which they are graduated outward ‘n degrees to 90. A measuring-arm, with a spatulate lower end beveled to receive vernier graduations, sweeps these scales. This arm is carried by a short shaft journaled in the upper edge of the plate and concentric with the two curved scales. The outer end of the shaft is furnished with a nurled hand-wheel, by which the clinometer is operated. The inner end of the shaft carries a frame which supports the tubular case containing the spirit- tube of the hand-level. The center of the case is cut away to show the bubble in the tube. On the extreme inner end of the shaft is a jam for setting the instrument, which, when turned inward, holds the shaft against turning. The measuring-arm and frame are rigid with the shaft. The case stands at a right angle to the measuring-arm, so that when the arm is placed at the zero-point of the two scales the case will be exactly parallel to the longitudinal axis of the hand-level. Immediately below the exposed portion of the spirit-tube a slot is cut in the top of the main tube. A small mirror is fixed DETERMINATION OF THE HEIGHT OF STANDING TREES, 147 at an angle inside the main tube directly underneath the slot. This mirror is so narrow and is placed so close to the side of the main tube that it does not obstruct the line of vision through the tube. ‘The observer can thus see at the same time the cross- wires at the objective and the reflection in the mirror of the spirit- bubble. In measuring the height of a tree the observer sights the instrument at the tip and turns the hand-wheel until the bubble shows that the case is level. The measuring-arm, which swings with the case, then indicates upon the right-hand scale in degrees an angle formed by a line running from the observer’s eye to the top of the tree and a horizontal line extending from his eye to the trunk of the tree. He then consults a table of natural tan- gents, which gives him the value of the angle secured, expressed as its tangent or percentage. The tangent or percentage of this angle is multiplied by the horizontal distance from the observer’s eye. He then sights to the base of the tree, and in thesame manner ascertains the angle formed by a horizontal line running from him to the tree and a line running from his eye to the base of the tree. He now consults his table again for the value of: this angle expressed as its tangent or percentage, and multiplies this value by his horizontal distance from the tree, which gives the height of the tree from the ground to the level of his eye. The figures thus secured are added Mpg te giving the total height of the tree. The scales of the instrument are sometimes graduated in tangents or percentages of angles instead of in cic in which case the table of tangents is not needed. 75. Other Height Measures.—Below are listed those other instruments which, by reason of cost, rarity, or serious disad- vantages for practical use, seem to the author unlikely to be used in the United States. Students who may wish to make a special study of these instruments are recommended to consult the German books on Forest ae Sa notably Miiller’s Holzmesskunde. : 145 FOREST MENSURATION,. 1. Rueprecht’s Height Measure. Sold by A. Rueprecht, Vienna, Favoritenstrasse 25. Price 50 Crowns. 2. Havlick’s Staff Height Measure. Sold by Gebr. Fromme, Vienna. Price 32 Crowns. | 3. Havlick’s Hand Height Measure. Sold by Gebr. Fromme, Vienna. Price 9 Crowns. 4. Fricke’s Nasenkreuz Height Measure. Sold by Mechaniker Fentzloff, Hann.-Miinden. Price 1.50 Marks. 5. Stélzer’s Height Measure. Sold by E. Bischoff, Meinin- gen. Price 40 Marks. 6. Ed. Heyer’s Height Measure. Sold by W. Sporhase, Giessen. Price go Marks. 7. Triimbach’s Height Measure. Sold by Trimbach, Kgl. Bayr. Forstamtsass’t, Obernburg a. M. Price 90 Marks. 8. The Omnimeter. Sold by Keuffel & Esser Co., N. Y. Price $15. 9. Bose’s Height Measure. Sold by Mechaniker Weingarten, Darmstadt, Germany. Price 6 Marks. 10. Mayer’s Height Measure. Sold by L. Tesdorpf, Stutt- gart. Price 45 Marks. 11. Tiemann’s Height Measure. Must be made to order. Keuffel & Esser Co., N. Y. Price about $50. The dendrometers described in section 78 may all be used as height measures. 76. General Directions for the Measurement of Heights of Trees.—It is important to select for observation a station whence both the tip and the base of the tree may be distinctly seen. The tip of -a coniferous tree is easily distinguished; but on a hard- wood with a round top it is often difficult to distinguish the true tip from a side branch. If one should mistake a side branch on the near side for the tip, the measured height would be too large, The true tip may usually be seen by standing at some distance from the tree. The general rule is to choose an observation station at a distance*approximately equal to the height of the tree; or if this is impractical, at a greater rather than a less dis- DETERMINATION OF THE HEIGHT OF STANDING TREES. 149 tance. On a slope one should stand above rather than below the base of the tree. It is often difficult to distinguish the true base of a tree on account of brush, grass, or other obstacles. It is necessary to get a clear vision to the base; and if it cannot then be easily seen, a handkerchief or some other bright object may be used as a distinct point for sighting. If a tree is leaning, one should choose a position in a line perpendicular to the vertical plane of the tree. The result of the measurement is then the distance from the ground to the tip of the tree and not the length of the tree itself. To determine the exact length of the stem of a leaning tree one would have to calculate the degree of inclination and from this reckon by trigo- nometry the true length. The object of taking a position at right angles to the direction of inclination is to avoid a possible mistake in measuring the distance from the tree. If one were in line with the lean of the tree and measured the exact distance to the base a false result would be obtained. With most instruments the horizontal distance from the observer to the tree is required. In the case of the Klaussner hypsometer the oblique distance from the eye to the base of the tree and not the horizontal distance is measured. In all cases the distance should be measured, not paced. In using the Klaussner instrument a beginner sometimes has difficulty with a leaning tree whose tip is not in vertical line with the base. The best plan is to sight to an imaginary point on a level with the true tip. If the instrument were per- fectly leveled it would be possible first to take an observation to the base of the tree and then to turn the instrument enough to sight the tip. There is, however, no way of leveling the Klauss- ner height measure, and therefore the first method is the best. 77. Choice of a Height Measure.—Inasmuch as the choice of twe equally good instruments depends on individual taste, opin- ions differ among foresters as to the best height measure. In the opinion of the author the Klaussner height measure is the best 150 FOREST MENSURATION. of the small instruments for accurate scientific work. For general forest work where accuracy is desirable, but great precision is not necessary, the Faustmann height measure gives the most satisfactory results. 78. The Use of Dendrometers—A considerable number of instruments have been manufactured for measuring the diameter of standing trees. at different points. Several of these instru- ments are accurate, and may be used to great advantage in scien- tific work. They will probably not be much used in general forest work, for the following reasons: first, a good instrument is rather expensive; second, most dendrometers are delicate and easily thrown out of adjustment; third, as a rule they are rather large and cumbersome to carry, in addition to requiring a tri- pod; fourth, an observation, if properly made, is time-consuming; fifth, only the diameter outside the bark can be measured directly, the inside measurement being that usually required. The dendrometer of Winkler described in section 71 is, in the judgment of the author, the best instrument for foresters from the standpoint of simplicity of construction, compactness, rapidity of measurement, and cost. The Wimminaur dendrometer has been recommended by several American authors. It is, however, much more expensive than the Winkler instrument, and, on account of the delicate construction of certain parts, is less adapted to rough forest — work. It may be purchased from W. Sporhase, Giessen, Ger- many, for about 75 marks. The best instruments for purely scientific work are those of Joseph Friedrich, head of the experiment station at Maria- brunn, Austria, and of A. R. von Guttenberg, of the forest school in Vienna. These are instruments of great precision, and may be used in taking periodic measurements in studying diameter growth. They have to be made to order, and are therefore very expensive. Other dendrometers are: Peltzman’s dendrometer, which can be secured from Gebr. Fromme, Vienna, price in Vienna 80 crowns; Starke’s dendrometer, sold by Starke & Kam- DETERMINATION OF THE HEIGHT OF STANDING TREES, 151 merer, Vienna; Sanlaville’s dendrometer, sold by Neuhofer & Sohn, Vienna, price 70 crowns. A special attachment for measuring diameters is made for the Klaussner height measure. lor a detailed discussion of the different dendrometers, see Miiller’s Holzmesskunde. CHAPTER XI. DETERMINATION OF THE CONTENTS OF STANDING TREES, 79. Estimate by the Eye.—Persons who have constant prac- tice in measuring logs and trees are able to estimate the contents of standing trees by a mere superficial inspection. Practiced timber cruisers attain an astonishing degree of accuracy in such estimates. The estimating of contents of trees at a glance is possible only by a trained eye. The inexperienced cruiser or one who is estimating an unfamiliar species must calculate the contents of standing trees from measured or estimated diameters and by the use of a log rule. It is necessary first to determine the lengths of the logs; then the diameter inside the bark at the top of each log. The scale of each log is obtained from a log rule and és results for the different logs added together for the total scale of the tree. This method involves the ability to estimate diameters at different points up the tree and involves also a knowledge of the thickness of the bark, which varies at different points. For one not practiced in estimating diameters, a good method is to use a light ro-foot pole, attaching across one end a small stick marked off in inches by prominent notches. An assistant holds the pole against the trunk with the cross-stick at the point where the top of the first log would come. The diameter can be de- termined very accurately from the notched rule. From a diam- eter measurement at the base, taken with calipers, and another at the top of the first log obtained as just explained, it is possible 152 DETERMINATION OF THE CONTENTS OF STANDING TREES. 153 to estimate by comparison the diameters at the tops of the other logs. ‘The average thickness of the bark at different heights must be allowed for. After one has taken a few measurements of bark on felled trees, it is possible to estimate fairly well by its general appearance the thickness of bark on standing trees. After sufficient practice in estimating diameters, the 1o-foot pole may be discarded. This method of determining contents works extremely well with trees having not over three or four logs. The butt log, since it has the largest scale, requires the most accurate estimate. If the contents of this log is correctly determined, a slight error in estimating the diameters of the others will not materially affect the total scale; except, of course, in tall trees with more than four logs. Naturally the work of estimating the diameters can be done more accurately with a good dendrometer. The time consumed in using a dendrometer, however, is so great that the method just described is more applicable to our con- ditions. One method often used is to estimate the length of the mer- chantable portion of the tree, then estimate its top and base diameters, average these diameters, and determine the con- tents by the Doyle rule. If the length of the merchantable por- tion of a tree is 4o feet, the top diameter 6 inches, and the base diameter 14 inches, the average diameter would be assumed to be ro inches, and the volume of the log would be, by the Doyle rule, go board-feet. A number of rules of thumb are in existence for estimating the number of board-feet in standing trees. The following is a good illustration: Subtract 60 from the square of the estimated diameter at the middle of the merchantable length of the tree, multiply by o.8, and the result is the contents in board-feet of the average log in the tree; multiply by the number of 16-foot logs for the total scale. A rule of thumb proposed by Dr. Schenck for estimating tall 154 FOREST MENSURATION. sound trees by the Doyle rule is: to square the diameter, breast- high, multiply by 3, and divide the result by 2. Still another rough method of Dr. Schenck is as follows: Assuming that the tree is to be cut into 16-foot logs and the taper is 2 inches per log, multiply the breast-height diameter of the butt log inside bark, by the number of logs, and multiply the result by the same diameter less 12. A quick and fairly accurate method of estimating the volume of white pine in cubic feet is as follows: Square the breast-height diameter (in feet) and multiply by 30. The rule gives excellent results for trees 10 to 14 inches in diameter and 8o feet high, 16 to 20 inches by 85 feet, 22 to 28 inches by go feet, and 30 to 36 inches by gs feet. For other heights add or subtract for each 5 feet of length 6% for trees 10 to 20 inches, 5.5% for trees 22 to 28 inches, and 5% for trees 30 to 36 inches in diameter. Suppose, for example, a tree is 18 inches in diameter and go feet high, the rule would be: Square 1.5, multiply by 30, and increase the result by 5.5%, or (1.5)?X30+5.5%=71.2 cubic feet. The volume table in the Appendix gives 71.9 cubic feet for such a tree. A similar rule * is used in Germany to obtain the volume of standing trees in cubic meters, as follows: Square the diameter, breast-high, in centimeters, and divide by tooo. The rule holds good for pie 30 meters high, beech, oak, and spruce 26 meters high, and fir 25 meters high. For other heights add or subtract the following amounts for each meter of length, accord- ing as the tree is taller or shorter than the above heights: Pine, add 3%, subtract 3%. Beech,... ** 5% + We Spruce, 1°. 3% > 4%. Fir, ae a 5 * Called Denzin’s method. DETERMINATION OF THE CONTENTS OF STANDING TREES. 155 80. Estimate of the Contents of Standing Trees by Volume Tables and Form Factors.—Volume tables are used by foresters in this country more extensively than any other method of esti- mating the contents of standing trees. In Europe the method of form factors, as well as that of volume tables, is used. These methods are described in the next chapters. 81. Rough Method of Estimating the Cubic Contents of Standing Trees.—The cubic contents of the stem of a tree may roughly be obtained from the measured diameter at breast- height by the formula igs 2 ? in which V is the volume, B the area at breast-height, and H the height. By reference to page 88, it will be seen that this is for- mula No. 3 for cubing a paraboloid. 82. Hossfeldt’s Method.—On page 94 it was shown that a log may be cubed by the formula, h in which V is the volume, B, the sectional area at one-third the distance from the butt, 6 the sectional area at the top, and h the length of the log. In case of an entire stem 0 iso, and the formula becomes V =< $B, x h. To determine the cubic contents of a standing tree, the length of the stem above the probable stump is measured with a height measure, and the diameter at 4 this length is estimated or is measured with a dendrometer. These measurements furnish data for the application of the Hossfeldt formula. 83. Pressler’s Method.—In 1855, G. Pressler, a professor in in the Forest School in Tharandt, devised the following formula for cubing a standing tree: 3 2 156 FOREST MENSURATION. in which V is the volume of the tree, B is the sectional area meas" ured just above the butt swelling, H is the distance from the stump to the point on the stem where the diameter is exactly one-half that measured at the butt, and M is the distance from the stump to the point where B is measured. Ordinarily B is taken at breast-height. The stem of a tree is cubed as two sec- tions, (1) the portion above the point where the diameter is taken, considered as a paraboloid or a cone; (2) the portion between the stump and the point of diameter measurement, considered as a cylinder with a diameter equal to that at the upper end of the section. The stump and branches are disregarded. The main part of the stem is cubed by the formula, V=3BXxh, in which h is the distance from B to the one-half diameter-point. This holds good for both the paraboloid and the cone, as may be seen in the following demonstration: In a paraboloid the point at which the diameter is } that at the base, is 3 the altitude. If this distance is /, the total altitude A , the basal area B, then h=3H, and H=$h — Substituting in the formula BH | Vee. lak Sy ae 2 2 3 3 The same process of reasoning will show the formula correct also for the cone. The lower part of the tree is cubed as a cylinder by the formula, V=B M. The volume of the whole stem is then V=2Bh+BM DETERMINATION OF THE CONTENTS OF STANDING TREES. 157 A dendrometer may be used to determine the point where the diameter is one-half that at the base. Pressler devised a special instrument for this purpose, consisting of a small paste- board telescope with an eyepiece at one end and with two pins or screws at right angles to the axis of the instrument at the other end. In use the telescope is first closed and the tree is sighted where the base diameter is to be taken. The pins or screws are adjusted so that the stem appears to occupy the space between the points. The instrument is then drawn out to twice its former length, and sighted up and down the stem to find the point exactly fitting between the pins. The diameter of the tree at this point is one-half that at the point first measured. CHAPTER XII. VOLUME TABLES. 84. Definition of Volume Tables.—Volume tables show the average contents of standing trees of different sizes. They may be made for any desired unit—the cubic foot, board foot, or they may show the contents of trees in ties, poles, shingles, or other product. They are standard, cord, or cubic meter used to estimate the yield of wood and timber standing on speci- fied tracts. Volume tables are intended only for estimating a large number of trees. Compiled from the average of a number of measurements, they are necessarily inaccurate as applied to a single tree. ‘The volumes of individual trees of the same species and same dimensions may vary 20 percent. or more. On the other hand, the average volume of a large number of trees of the same species, having the same height and diameter and growing under the same conditions, is very uniform; and tables showing the average volumes of a large number of felled trees give satis- factory results in estimating the contents of a large number of standing trees. Volume tables may be local or general. Local volume tables are based on the measurement of trees growing in a restricted locality and usually under specified conditions of mixture, density, etc. General volume tables are based on the average volume of trees growing in a variety of conditions over a large region. In Germany general volume tables are usually made. In this country the forests are so irregular in age, density, and form that 158 VOLUME TABLES. 159 local volume tables are the rule; and often there must be separate tables for areas as small as townships or counties. The best rule is to make separate tables at least for every forest region. 85. Volume Tables for Trees of Different Diameters.—The simplest and most convenient volume tables show the average contents of trees of different diameters. ‘These are the tables in most common use in estimating the merchantable contents of standing timber. The total contents of trees of any given diam- eter are computed by multiplying the number of trees by the average volume given in the volume table for that diameter. (See page 219.) The tables are based on the measurement and computation of volume of a large number of felled trees. These data are usually secured where lumbering is in progress. A crew of two or three men follow the cutters and measure the trees as they are felled. If the investigation includes only the preparation of volume tables, the following measurements are usually taken on each tree: diameter at breast-height; diameter at each cross-section, inside and outside the bark; length of each log; length of the top above last cut, and height of stump, giving total height; length of crown, and width of crown. With these measurements, the merchantable or full contents of the stem, with and without bark, may be computed. The measurements of the length and width of crown serve as an excellent description of the tree. In addi- tion, it is usually desirable to add a descriptive note regarding the form of the trunk, soundness, general thrift, approximate age, the form of the stand, the trees in mixture, and the soil and situa- tion. Where a volume table is constructed for diameters alone, a full description of the tree and forest is not essential. A crew of three men is most effective for work in collecting measurements of volume. One man selects the trees, directs the work, and records the measurements. The others do the measuring. The proper equipment of such a crew is a pair of calipers, tape, scale rule, and record book or tally-sheets. Before undertaking the field work of collecting material for | 160 FOREST MENSURATION. volume tables, it is desirable to examine the forest where the tables are to be used in estimating, in order to determine what type of trees ought to be measured. It is then decided how many trees to fell and measure, and in general how they should be distributed among the different diameters. Ordinarily the aim is to measure at least 1000 trees as a basis for volume tables; but where the timber is very uniform, as with most conifers, 500 trees give exceedingly good results. If the tables are to be used in careful cruising, at least 500 trees should be measured. In reconnaissance work and rough cruising, or where the trees are extremely regular in form, 100 trees may suffice. Care is required in the selection of the trees for measurement. It is the rule to measure only sound trees, because volume tables show the full contents of sound trees. It might appear that the tables would be more practical if based on average trees, including those partially defective. But a table made up in this way would be extremely unreliable, for it is well known that the defects of trees differ greatly in different situations; so that a table based partly on defective trees would be useless in estimating trees whose defects are different from those of the trees observed in its construction. Again, any such defect as injury by fire, insects disease, wind, or ice, would entirely vitiate a table constructed for trees showing another defect than the particular one in ques- tion; whereas a table based on sound trees may be reduced in any given case, just as log rules are reduced for unsoundness in logs. Care should be exercised to select for measurement trees representative in form. The temptation usually is to measure only the best trees; but it must be remembered that the figures will represent the average tree of each diameter, regardless of difference in the number of logs, total height, or tree class. There- fore each tree should be a good representative of its class, and normal in height, size of crown, form of trunk, etc. Different classes of trees should be represented about as they occur in the forest; that is, there should be about the same percentage of VOLUME TABLES. 161 one-log trees, two-log trees, three-log trees, etc., as ordinarily occur in the particular forest under observation. This point is to be especially observed when the number of trees measured is limited. If 1000 trees are to be measured, it is ordinarily suffi- cient to measure trees as they are cut by the lumbermen, taking care that the diameters are well distributed and that the trees are not abnormal. Abnormal trees are those with forked trunks, those with swollen butts, and diseased or distorted trees. The first work of computation is the calculation of the volume of each tree measured. The work can be done most rapidly by two persons, one handling the data collected in the field and the other the log tables, tables of areas, or other tables neces- sary in determining the contents of the logs in the unit chosen. The computing work may, with economy of time and mental effort, be divided between the two. The trees measured are grouped according to breast-height diameters in inch classes. Thus the 6-inch class comprises all trees with a diameter between 5.6 and 6.5 inches. In judging the diameter class the five-tenths goes to the lower rather than the higher number; that is, a tree 12.5 inches in diameter is counted as a 12-inch tree, not a 13-inch tree. The volumes of all trees in a single-diameter class are averaged together and the exact average diameter also determined, the last being usually not a whole inch, but a few tenths, above or below the whole number. The data may then be arranged in five columns, as in the table on page 162. The first column shows the inch-diameter classes, the second column the exact average diameters of the trees in each diameter class, the third column the number of trees used, the fourth column the average volume of the trees in each diameter class, and the fifth column the results of the fourth column made regular by graphic interpolation. To construct the curve used in obtaining the values in column five, the volumes from column four are plotted on cross-section paper as ordinates, with the average diameters in column two as abscisse. The values in column three show what points are to receive most em- 162 FOREST MENSURATION. phasis in drawing the curve. For the final results in column five, the values for the whole inches are read from the curve. CHESTNUT—VOLUME TABLE FOR TREES OF DIFFERENT DIAMETERS. BASED ON THE MEASUREMENT OF 101 TREES AT MILFORD, PA. Average | Average Volume Diameter Class, Diameter of | ; Average Volitme Results ot Breast-high, | Trees Measured, Number of ot Trees _ Column 4 Teaches. Breast-high, Trees Measured. Measured, Evened Off by Inches. Cubic Feet. Curve, Cubic eet. 6 6.25 2 4.7 4.5 7 7 10 5-4 5-4 8 8.1 13 7.9 7 9 9 16 9.4 9.3 10 10 15 ee E39 II II 14 14.9 14.4 12 12 18 16.2 17 13 12.9 9 20.2 20.2 14 14.1 3 23.6 23-4 15 14.9 I 27 27 ‘ i Another method of averaging together the volumes for different diameters is as follows: The volumes of all trees are plotted on cross-section paper as ordinates, the abscisse being the diameters breast-high. After the volumes of all trees have been plotted, an average curve is drawn through the points. From this curve are read the average volumes for the different diameters. Volume tables for trees of different diameters give very satis- factory results in cruising. particularly when the tables have been prepared for special conditions. They are not, however, applicable to forests where the conditions of growth differ from those pre- vailing where the data were gathered. If, for example, the tables were based largely on tall trees, they could not be used where the trees are, On an average, shorter. This objection is largely obviated by making local tables for restricted areas, on which the general conditions for growth are fairly uniform. Volume tables for trees grouped by diameters alone are de- signed primarily for commercial estimating in board measure. VOLUME TABLES, 163 A further grouping of the trees is necessary for very close deter- mination of volume, as described in the succeeding sections. 86. Volume Tables for Trees Grouped by Diameter and Number of Logs.—In the method just described all trees are averaged by diameters regardless of height or length of mer- chantable timber. Thus one-log trees are averaged with three- log trees, or even five-log trees, of the same diameter. In order to secure greater accuracy, volume tables based on trees grouped by diameters and number of logs were devised. Such tables are in actual use by cruisers. They are used in tall timber where a standard log-length—as, for example, 16 feet—may be used in the estimate of the number of logs. To construct a volume table for trees grouped by diameters and number of logs, a large number of felled trees are measured and their volumes computed as described for the previous method. The trees having the same number of logs are then grouped together, and the average volumes of one-log trees of different diameters are determined, then of two-log, three-log trees, etc. If the volumes do not increase regularly with increase of diameters, the irregularities are evened off by graphic interpolation. The results are expressed in a form like the following: VOLUME TABLE FOR TREES OF DIFFERENT DIAMETERS AND NUMBER OF LOGS. LENGTH OF STANDARD LOG...... FEET. (Based on the Measurement f...... Trees.) 7 ul Diameter, Volume of Volume of Volume of Volume of Breast-high, One-iog Trees, Two-log Trees, | Three-log Trees, | Four-log Trees, Inches. Board Feet. Board Feet. Board Feet. Board Feet. The objection to this method is that trees are not always cut into logs of the same length. Even with very tall trees it is seldom that all the logs are the same length. A tall white pine may 164 FOREST MENSURATION. for example, yield three-sixteens, and one twelve. If the volume tables are based on sixteen-foot logs, an inaccurate estimate would result if this were classed as a four-log tree. ‘These tables are, therefore, not much more accurate than those first described. 87. Volume Tables for Trees of Different Diameters and Mer- chantable Lengths.—On account of the defects of the previous method, it has been proposed to base volume tables on trees grouped by merchantable length as well as by diameter. The length classes should be such as would be actually used in practice. When short logs are used, the merchantable length of a given tree would be the sum of the log lengths. In this case the length classes would have to correspond to all the possible combinations of short logs. ‘Two-foot classes would meet this requirement, and would also be small enough for the conditions where the whole merchantable part of the tree is taken out as one log. To construct such a table, the measurements of felled trees are first obtained in the ordinary way. The computed volumes are grouped by the diameters and merchantable lengths of the trees, each length class comprising two feet. A preliminary table of averages is made, giving the average volume of trees of different diameters with a merchantable length of 10 feet, those with a merchantable length of 12 feet, those with 14, 16, 18, 20, 22 feet, and so on. Under ordinary circumstances it will be found that this table has irregularities not only in the vertical columns, but also in the horizontal lines. These irregularities are then evened off by a series of harmonized curves. The final table may be expressed in a form like that shown on page 165. As far as the author is informed, no such volume eid have been made. They should, however, yield very accurate results, although more difficult in application than the volume tables based on diameters alone. Their use in estimating stands is described on page 222. 88. Volume Tables for Trees of Different Diameters and Tree Classes.—These are designed for use where the trees have grown under varying conditions of density and form of the stand, — — ——__ eee VOLUME TABLES. 165 VOLUME TABLE FOR TREES OF DIFFERENT ‘DIAMETERS AND MERCHANTABLE LENGTHS. BASED ON THE MEASUREMENT OF...... TREES. Merchantable Length in Feet. piers 10 | 12 | 14 | 16 | 18 | 20 | 24 | 28 | 30 Inches. Board Feet. oS as in very irregular forests. Such tables are particularly useful in estimating cord-wood in second-growth hardwood forests. The author has found that volume tables based on diameter alone are not accurate for cordwood work, when large branches are merchantable, a table based on merchantable length is out of the question. On the other hand, a table which gives sepa- rately the volume of the trees with large crowns, those with medium crowns, etc., yields very good results. No rules can be laid down for the formation of classes. Under some circumstances, it might be desirable to make three classes—dominant, interme- diate, and suppressed trees. Elsewhere a grouping of trees with large crowns, medium crowns, or small crowns would be proper. In second-growth hardwoods the following classifica- tion will be found to be useful: 1. Trees in the open. 2. Large-crowned forest trees (maximum in stand). 3. Trees in crowded stand, crowns narrow and about 15-20 percent of the length of stem. 4. Overtopped and partially suppressed trees. 5. Badly suppressed trees. In selecting the trees for volume measurement, much greater stress is placed on the description of the trees than with the other kind of volume tables. It is particularly important to describe 166 FOREST MENSURATION,. the conditions of ‘density, form of surrounding stand, and shape and dimensions of the crown, because these are the factors which determine the class to which a particular tree is assigned. After computing the contents of the trees, they are separated into classes, and then for each class a table is constructed in the ordinary manner, showing the volume of trees of different diameters. These separate tables are then combined in the following form: VOLUME TABLE FOR TREES OF DIFFERENT DIAMETERS AND TREE CLASSES. BASED ON THE MEASUREMENT OF...... TREES. Tree Class. ; Diameter Breast-high, I Il III IV V Inches. Cords. 89. Volume Tables for Trees of Different Diameters and Heights.—These are usually considered the most accurate kind of volume tables. The European volume tables, which are used with satisfactory results, even where considerable accuracy is required, are based on this principle. In Europe, however, the ordinary volume tables are used in estimating regular forests and separate tables are made for different age classes. Even when used in very irregular stands, where the trees differ largely in age and development of crown, they are more accurate than volume tables based on diameter alone. They are probably not as accurate as those based on diameter and merchantable length, because the merchantable length is a better index of the volume of a tree of a given diameter than the total height. Volume tables based on diameter and height have been constructed for several species in this country and used in prac- tical work of estimating. They give good results with trees of VOLUME TABLES, 167 regular form like the pines and spruces, but with the hardwoods they are not entirely satisfactory unless separate tables are made for different tree classes. The construction of volume tables for trees of different diame- ters and heights is based on the measurement and computation of volume of a large number of felled trees. European volume tables are based on tables of form factors. In this country a number of tables have been constructed by averaging together directly the volumes of the measured trees, grouped by diameters and height classes. ‘The procedure in this method is as follows: The computed volumes of the measured trees are grouped by inch diameters and five-foot height classes. Figures of each diameter and height class are then averaged together and the results compared in a preliminary table. This preliminary table, even if based on a very large number of measurements, usually is irregular in both the horizontal and vertical directions. All values are then evened off by a series of harmonized curves. The final form of the table is illustrated by the following example: CHESTNUT—VOLUME TABLE FOR TREES OF DIFFERENT DIAMETERS AND HEIGHTS. BASED ON 99 TREES MEASURED AT MILFORD, PA. Height in Feet. Diameter, | | Breast-high, 40 45 50 55 60 Inches. Merchantable Cubic Feet. 6 476 Ai2 4.6 7 4.8 5-1 5-7 8 6.2 6.6 Soe, | 8.1 9 7.8 S3 9.0 10.0 10 ee 10.2 i De 2.2 II 12.0 12.6 13.6 14.8 16.3 72. Np evare eee 5.x 16.1 17.4 18.9 E53. °. WS. ahaee ee 17.8 18.9 20.2 3 ay | Teh 1) Ee de tee 20.9 22.0 23.4 25.0 ES: ' ¢ boca ates eee Tene ee Se 28.8 168 FOREST MENSURATION,. Another method, and the one most commonly used in Europe, is first to make a table of form factors and then convert this into a volume table by multiplying each value by the product of the corresponding height and basal area. ‘The conversion of a form- factor table into a volume table should present no difficulty to the student after reading the next chapter, which describes the theory and use of form factors. This last method is applicable only to cubic measure. Often it is desirable to make a volume table in cubic feet as a founda- tion for a table in some other unit. One of the best ways, from the standpoint of accuracy, of making a volume table for mer- chantable timber is first to construct a table of cubic contents of the entire stems of trees and then reduce this to a table of merchantable contents. It is a good method because the con- tents of whole stems do not vary so much as the merchantable volume and a table of averages may be constructed with less interpolation. A table showing the ratio of merchantable to total contents may then be constructed and applied to the first volume table to reduce the values to merchantable terms. Suppose that a table of cubic contents of the stems of white pine has been constructed, like that in the Appendix, and one wishes to convert it to board feet, the procedure is as follows: The volume of each of the trees measured is computed in cubic feet and also in board feet and the ratio between the two determined. The cubic volume in each tree is multiplied by 12 and the board feet divided by this product. The result represents an artificial but convenient ratio between the cubic and board feet of each tree. A table of factors is then constructed for trees grouped by diameters and heights or by diameters alone. Such a table was made by the author in constructing volume tables for white | pine, and may be used as an illustration (see page 169). The cubic volume table is then converted into board measure. Each value is multiplied by the factor in the table corresponding to the diameter, and the result multiplied by 12 in order to con- vert back to board feet. VOLUME TABLES. 169 RATIO BETWEEN THE BOARD CONTENTS AND TOTAL VOLUME OF WHITE PINE. Board Feet pach shed Ses Feet Reduced to Reduced to Reduced to : Cubic Feet in ‘eerie ge Cubic Feet in ane Cubic Feet in Diameter, Percentage Diameter, Percentage Diameter, Percentage Breast-high, ra Breast-high, Breast-high, ree Tan of the Total niches of the Total Tante of the Total ah raat Volume of eee Volume of TR Volume of Wood and Wood and Wood and Bark. Bark. Bark. 10 12 22 35 34 46 I2 18 24 38 36 47 14 23 26 40 38 48 16 26 28 42 40 49 18 29 30 44 20 32 32 45 go. Graded Volume Tables.—The volume tables described in the preceding pages give the contents of trees in a given unit, but they do not indicate the quality of the product. Graded volume tables show for trees of different sizes the amount of timber of different grades and enable the determination of the money value of standing trees better than the ordinary volume tables. The U. S. Forest Service is at present studying the yield, in timber of different grades, of all the important trees. Already investigations of long-leaf pine, loblolly pine, yellow poplar, white oak, chestnut, ash, and other hardwoods in the South, and maple, birch, and beech in the Adirondacks, have been initiated. The results of such studies are expressed first as graded volume tables, and second as tables showing the money value of trees of different sizes. | The method of constructing a graded volume table is as follows:* A large number of trees are measured as soon as they are cut by the saw crews. The length and top diameter of each log are measured and a mark placed on the end. Each tree is given a number and each log in that tree an additional number or letter. Thus, for example, 576? indicates the second log * From the Determination of Timber Values, by E. A. Braniff, Year book of the U. S. Dept. of Agriculture for 1904. 170 FOREST MENSURATION. (counting from the butt) of tree 576. The logs are then sawed at the mill and their exact yield in graded lumber determined. The measurements at the mill are taken in the following way: A man stands near the slab-carrier. As each piece of siding from a marked log is dropped on the rollers, the number of the log is chalked on it. When a siding has passed through the edger and trimmer, it is graded and measured and a record made of the log number, grade, and dimensions. These data for all boards cut from the marked logs enable the computation of the exact product in graded lumber of each log, and by com- bining the products of all logs having the same tree number, the total yield of each tree is computed. The data taken in the woods serve a check on the work at the mill. When the volume of all the trees has been determined, the trees are grouped together by inch-diameter classes and the average contents of trees of each diameter computed. Any irregularities in the resulting table are evened off by curves. The form of these tables is well illustrated by the Graded Volume Table for Yellow Birch as shown on page 171. This is the result of an investigation in the Adirondacks by the U. S. Forest Service. This table shows the yield of choice grades of birch advancing rapidly with the growth of the tree. The choice grades are firsts and seconds red, and firsts and seconds. The amount of red birch in a tree under 18 inches in diameter is too small. to consider. An 18-inch tree contained 2 board feet of this high- priced lumber, a 19-inch tree only 4 feet of it, a 20-inch tree 8 feet, but in a 21-inch tree the amount rose to 23 board feet, showing a gain of almost 200 percent. over the product of the previous diameter. The explanation for the exceptional increase is that the rules of the National Hardwood Lumber Association, under which the lumber was inspected, require red birch 4 or 5 inches wide to show one face all red; over 5 inches, one face must be not less than 75 percent red. Red birch is heart-wood, and it happens that the heart-wood is not wide enough to pass the VOLUME TABLES. 171 GRADED VOLUME TABLE FOR YELLOW BIRCH. Diam- | Firsts . Shipping |Mill Culls| Sound eter, ; and _ : No. 1 |Cul s( No. (No. 3 7 x9" Total. nh aoe eet fiomeog | Saconds,. |U Pome Mareen | rh See ee Talhied. Inches. | Bd. Ft. | Bd. Ft. | Bd. Ft. | Bd. Ft. | Bd. Ft. * Bd. Ft. SAA ks aise 3 5 6 20 25 59 7 ee eee 7 7 7 37 37 95 16 Sse Aiba ven ia h- II 10 8 4! 55 125 23 16 cde aii $5 16 12 8 38 72 146 32 hy ead | a Sree 22 14 8 35 84 163 32 18 2 28 17 9 36 94 186 ay 19 4 36 20 10 45 102 217 50 20 8 44 24 II 55 108 250 39 21 23 54 28 13 65 114 297 40 22 26 66 31 15 74 119 331 46 23 36 78 33 16 82 118 363 25 24 48 86 36 18 88 I12 388 37 25 62 g2 38 19 93 104 408 30 26 81 97 42 20 98 96 434 24 27 IOI 103 47 22 106 gI 470 28 28 116 110 53 22 118 86 505 16 29 128 120 59 23 134 81 545 4 30 139 132 64 24 155 74 588 12 31 150 144 68 25 180 52 619 4 * To obtain number of ties divide board feet in this column by 42. severe inspection in considerable quantities in trees under 21 inches in diameter. The increase of red birch goes on steadily from the 21-inch to the highest diameters. The next best grade, firsts and seconds, not graded by color, is contained in practically all sizes of merchantable trees. The increase of this grade goes on steadily, but is greatest between 18-inch and 23-inch trees, because the inspection rules, which favor wide boards, show their greatest effect here. Narrow boards from small trees grade lower than wide boards from large trees. When we compare the choice grades (firsts and seconds red, and firsts and seconds) with the common ones (No. 1 common, shipping culls, and mill culls) we find that the choice grades increase, on the whole, much more rapidly with the growth of the tree than do the latter. In the case of firsts and seconds red there was a rise between a 13-inch and a 31-inch tree from o to 150 feet, and in the case of firsts and seconds from 3 to 144 172 FOREST MENSURATION. feet. Contrast this with No. 1 common, which rises from 5 to 68 feet; with shipping culls, which rise from 6 to 25 feet; and with mill culls, which rise from 20 to 180 feet, and the tendency of the better grades to outstrip the poor ones becomes apparent. The fact must not be overlooked, however, that a considerable amount of what would have made inferior grades went, in this instance, into railroad ties. The table given on page 171 may be used to determine the money value of trees of different diameters. If the value of each grade of lumber is known, it is a simple matter to convert a graded volume table into a money value table. According to prices obtained from the Boston and New York markets for yellow birch, the table given above may be expressed as follows: VALUE OF YELLOW BIRCH. Diam- Di r, Graded Value lametety! Graded Va! Breast. Leppaed ae e. | Per 1000 a Volume. rae ae. dea" Inches. | Bd. Ft. Bd. Ft. Inches. Bd. Ft. Bd. Ft. 13 59 | $0.55 $9 . 32 23 363 $5.19 | $14.30 14 95 0.89 9.37 24 388 5.80 | 14.95 15 125 1.22 9.76 25 408 6.39 15.66 16 146 1,52 10.41 26 434 as 16.48 17 163 1.78 10.92 27 470 8.03 17.09 18 186 2.53 11.45 28 505 8.80 17.43 19 217 2.56 11.80 29 545 9:57 17.56 20 250 3.06 12.24 30 588 10.34 17.59 21 297 3.98 13.40 31 619 10.99 17.75 22 331 ae 13.63 This table illustrates how the value of trees increases with | increase of diameter. Not only do the trees increase in value, but their value per thousand feet increases as they grow larger. These tables may be used not only to determine much more closely than ordinarily the value of timber land, but when com- bined with a knowledge of growth they enable an owner to determine what trees should be cut and what should be left to accumulate increment. Graded volume tables must necessarily be made for restricted VOLUME TABLES 173 regions or localities. The methods of manufacture differ not only in different localities, but also at different mills. The quality of the timber varies in different regions, due to differences in the character of the soil, exposure, and other silvical factors. Graded volume tables should be made, therefore, to apply to a specified set of conditions. In the same way tables of timber values must be local in character, and in constructing them, one should consider the changes in price of the different grades. The graded volume tables now being made are based on diameter alone. It would, of course, be possible to make more elaborate tables for different tree classes, trees of different heights, etc. It is, however, proba- ble that the present method will answer the present needs of forestry. The principle of graded volume tables may be extended to comprise other products than lumber. Thus tables may be constructed to show the volume of chestnut trees of different diameters in ties of different grades, in posts, and in cord-wood. Volume tables may also be made for poles. Thus it would be of great practical value to have tables showing the average length and top diameter of poles yielded by chestnut of different diameters, or the length and middle diameter of piles contained in pitch pine trees of different sizes. rt} A 4\ Dwi! CHAPTER XIII. FORM FACTORS. or. Definition of Form Factors.—The form factor of a tree is the ratio between its volume and that of a cylinder having the same diameter and height. Expressed mathematically, =By V=BHP, in which F is the form factor, V the volume, B the basal area, and H the height of the tree. The form factor is, then, a reducing figure by which the volume of a cylinder having the same diame- ter and height as the tree must be multiplied to obtain the volume of the tree. Inasmuch as the tree has a smaller volume than the cylinder, this reducing figure is a fraction. It is customary to distinguish between breasi-height jorm jactor, absolute form factor, and normal form jactor. The breast- height form factor is obtained by the use, in the above formula, of the sectional area at breast-height; that is, the diameter of the cylinder to which the tree is compared is the same as the breast-height diameter of the tree, as shown in Fig. 33. The breast-height form factor is the one most commonly used, and in this book the term form factor will always mean bras height form factor unless otherwise indicated. If a tree is compared to a cylinder having a diameter equa] to the diameter at the base of the tree, the absolute form jactor results. According to the usual conception, the absolute form factor is based on a sectional area taken at the ground, as shown 174 FORM FACTORS. 175 in Fig. 34. If the volume used in the form-factor formula comprised only the portion of the tree above the stump, the sectional area would be taken at the stump in computing the absolute form factor. In the same way the absolute form factor mone ee SE E>=Ee Fa \ a ee ee es es ee ee eo is sometimes used to compare the portion of the stem above breast-height with a cylinder having a diameter equal to the breast-height diameter and having a height equal to the height of the tree minus 4.5 feet (Fig. 35). In this case the portion below breast-height is disregarded or. considered separately. Still another kind of form factor is obtained when the basal area is determined not at a fixed distance from the ground, but always at a distance having a fixed ratio to the height of the tree. The resulting form factor is called the normal or true form jactor. A still further classification of form factors is usually made. If, in the form-factor formula, the merchantable volume of the tree is used, the form factor is called the merchantable or timber jorm jactor. If the volume of the stem alone (including the top above merchantable wood) is used, the stem form factor is obtained. The tree form factor results when the entire volume of the tree, including branches, twigs, and all, is used in the formula. In this country only merchantable and stem form factors are used. 176 FOREST MENSURATION, 92. The Use of Form Factors.—Form factors are used to determine the contents of trees. If the basal area, height, and form factor of a tree are known, the volume is obtained by the formula V=BHF. ‘Tables are constructed showing the average breast-height form factors of trees of different dimen- sions, for use in estimating the cubic contents of standing timber. Just as volume tables, form-factor tables are designed for use in estimating the volume of a large number of trees and not of a single tree. European foresters constantly use form-factor tables in estimating standing timber. Convenient tables for different species are published in pocket handbooks like the Forst u. Jagd Kalendars of Prussia, Bavaria, and Austria. In this country form factors will be used chiefly in scientific work, as a basis for volume tables in cubic feet and other units, and as a method of comparing the form of trees of different species and of the same species growing under different conditions. They will not be used much in estimating timber because it is just as easy to construct volume tables, which are much more convenient for practical use in the field. 93. Variations in the Value of Breast-height Form Factors. —The term “ false’ form factor is sometimes applied to the breast-height form factor because it is not a true expression of form. This is best illustrated by comparing the breast-height form factors of two perfect bodies (as, for example, two A ppolo- nian paraboloids) which have different heights and different diameters. Inasmuch as ‘the two bodies have the same geo- metric form, the form factors, if true expressions of form, should be the same. But in our illustration the breast-height form factor of the paraboloid with the greater height is the smaller. ‘The reason for this is that the sectional areas are taken at the same distance from the ground, a point which in relation to total height is relatively higher on the shorter paraboloid. In consequence the volume of the cylinder to which the shorter paraboloid is compared is relatively smaller than in the case of the larger paraboloid, and the form factor is the larger. In FORM FACTORS. 177 other words, the breast-height form factors of trees of exactly the same form decrease with the increase of height. One would naturally conclude that tables of form factors would be constructed to show average values for trees of different heights without regard to the diameters. In fact such. tables were made in Europe—as, for example, those of Dr. Koenig—for Spruce and Fir.* But when the form factors of trees are com- pared, it is found that they do not always vary by the height alone, but sometimes by both height and diameter and some- times by diameter alone. The explanation of this lies in the fact that trees vary so much in form that the tendency of the form factors to decrease with increasing height is often counter- balanced. Influence of Density on the Size of Form Factors.—The forms of trees are distinctly influenced by the character of the forest in which they grow. It is a matter of common observation that trees growing in the open are apt to be more tapering than those growing in dense forests. Expressed in another way, forest trees are full-boled and the stem or merchantable form factor of a forest tree is ordinarily greater than that of a tree growing in the open. On the other hand, very old trees which stand in the open frequently have a greater stem form factor than those in the forest. In view of the variation in form factors due to differences in density of stands, it is desirable, particularly in this country where the forests are very irregular, to make separate form-factor tables for different tree classes. In 1864 the proposition was made by Dr. Koenig of Germany to make separate tables of form factors for the following classes of trees: First-class trees in crowded stand, slim and with narrow crowns. Second-class trees in stands of moderate density, more sturdy and wind-firm. * Hilfstabellen der Forstmathematik, by Dr. G. Koenig, 1864. 178 FOREST MENSURATION. Third-class trees in rather open stands, with full crowns. Fourth-class trees in open stands, with heavy crowns. Fifth-class trees standing singly. A similar classification is often desirable in this country. In much of our work, however, it may be better to make only three classes, as suggested for classifying trees in constructing volume tables (see page 165). Influence of Situation on the Size of Form Factors.—It has been long a matter of controversy whether the average form factors of a given tree vary in different localities and in different classes of soil. Without question the form factors are some- what influenced by the climate and by the character of the soil. Other factors, particularly density, are of much greater impor- tance, and usually trees from different forest types and diffreent classes of soil are averaged together in a single table. If the tree classes are kept separate, variations due to situations may be disregarded. Influence of Age on the Size of Form Factors.—Old trees have more cylindrical trunks than young ones. It is a saying among foresters that when a tree carries its diameter well up into the crown and has a full bole, it is old. This fact would indicate that the form factors increase with age. This is well illustrated in the case of the Norway spruce, whose stem form factor for trees 20 centimeters in diameter and over go years old is 0.559, but for trees under 90 years old and 20 centimeters in diameter is 0.534. Taking investigations of the different European trees as a whole we find no regular variation of stem form factors with age. This seems contra- dictory to the statement just made. But it is probable that only great differences in age affect the stem form factors and the trees used in the European investigations do not vary enough in age to be affected by this influence. Dr. Franz Baur of Ger- many proposes that separate tables of form factors be made for different age classes, each comprising 40 years. While it is a FORM FACTORS, 179 matter of contention whether differences in age have much influ- ence on the form factors of trees after they are go or 100 years is, til do certain that 4o-year age-classes are small enough. In our work with form factors it is desirable to make separate tables for young and for old trees. No rule can be given for the limits of the age-classes because these would differ under different circumstances. Thus it might be best to group the trees 50 to 100 years, or those 60 to 120 years of age. Ordinarily groups of 50 to 75 years will suffice for the present requirements of our investigations. Influence of Change in Diameter on the Size of Form Factors.— There is no uniform relation between form factors and diameters of trees. The form-factor tables given in the Appendix illustrate the lack of uniformity in the variation of form factors of the different species. 94. Construction of Tables of Form Factors.—TI'o construct a table of form factors a large number of trees are felled and measurements of volume taken. The volume and form factor of each tree are then computed and the results for trees of like diameter and height averaged together. Thus the form factors of all trees rounding to 6 inches and 25 feet are averaged together, the form factors of the trees 7 inches by 25 feet, those 8 inches by 25 feet, and so on. A convenient form for averaging the values is that on page 180, in which the number in the center of each square is the average form factor, the number in the upper left-hand corner is the average diameter of the trees measured, the number in the upper right-hand corner the number of trees averaged, and the number in the lower right-hand corner the average height of the trees measured. The next step is to inspect the values in this preliminary table to see how the form factors vary. If the form factors are found to vary regularly with the increase of diameter and height, it is necessary to make a table for both diameters and heights. If upon inspection of the first table of averages of the form factors, it is found that there is a distinct change in their values with 180 FOREST MENSURATION. Height in Feet. Diameter, a hae : Breast-high, 40 45 50 55 Inches. Average Merchantable Form Factors. 6— x 1°6:2 16,3 I 6 0.470 0.465 0.463 40 45-7 49 7-3 2|7 5 | 6.8 le 2 7 oO. 461 0.457 0.441 0.438 39-5 44.8 51.2 55 8.1 4 | 8.0 i ee % | 820 2 8 0.457 0.450 0.440 0.430 41 4 50 56 increase in diameter, but no appreciable change with increase in height, the final table is made by averaging together the form factors of trees of different diameters without regard to heights. In the same way, if the values of the form factors are found to change with increase in height, but not with increase in diameter, the form factor table is based on heights alone. It usually hap- pens, however, that there is a certain amount of irregularity in the horizontal and vertical column, even when the table is based on a large number of measurements. These irregularities may be evened off by a series of curves. . As a rule, tables of stem form factors for coniferous trees are based on diameter alone. Behm’s tables for Norway spruce, European fir, and larch over 90 years old are constructed on the basis of the average form factors of trees of different diameters without regard to height. The tables for silver fir and larch 60 to go years old are constructed in the same way, but the table for spruce 60 to 90 years old is based on both diameters and heights. The table of tree form factors for Scotch pine is constructed on a basis of heights without regard to diameters. Behm gives also a table of tree form factors for beech over go years old, based FORM FACTORS. 181 on diameters and heights, while that for beech 60 to go years old depends on heights alone. In spite of the opinion of many European authors that tables of stem form factors for mature coniferous trees should be based on diameters alone, the most important hand-books of Germany and Austria give, for use in practical estimating, stem form factors of these trees averaged by heights without regard to diameters. This lack of uniformity in construction of tables of form factors in Europe is extremely confusing to the student. It is best, in making a table of form factors, not to try to follow any rule, but to determine, by an inspection of the preliminary table of averages, whether the final tables should be based on diameters alone, heights alone, or both diameters and heights. Tables of form factors are considered worthless by European foresters unless founded on a very large number of measure- - ments, and, indeed, the form of individual trees varies so much that a satisfactory average cannot be obtained unless several thousand trees are measured. The Bavarian volume and form-factor tables, begun in 1846, were based on 40,000 trees. The tables for pine elaborated by Schwappach depend on 17,000. Baur’s spruce tables are based on 22,000 trees and Schuberg’s fir tables on 5643 trees. These German tables are designed to show the laws of the form of trees throughout large areas. This does not mean that in this country-the practical use of form factors is excluded under cir- cumstances where it is impossible to make such an extensive study. Local volume tables may be based on a relatively small number of measurements. The accuracy of such tables is, however, in direct proportion to the number of measurements used in their construction. But in the preliminary work of forest organization in this country the forester must often be satisfied with tables which give only approximate results. Local tables of volume and form factors based on roo trees are often used in this country with fairly satisfactory results. They must, 182 FOREST MENSURATION, of course, eventually be replaced by tables based on more thorough investigations. If the trees are separated into classes, 100 trees for each class will often suffice for volume and form-factor tables to be used in estimating standing timber. 95. Absolute Form Factor.—As explained above, the abso- lute form factor is obtained by dividing the volume of a tree by the product of its height and the basal area taken at the true base. Naturally the absolute form factor is not applicable to the full contents of a tree because of the difficulty of measuring the diameter at the ground. It is, however, sometimes applied to the portion of the tree above breast-height.* In this case the absolute form factor is found by dividing the volume of the tree above breast-height by the product of the breast-height sectional area and the height of the tree above that point. When this factor is used in cubing trees, the portion below breast-height is con- sidered a cylinder having the same diameter as that at breast- height. 96. Normal Form Factor.—This is a theoretical formula which was at one time used in practice by its inventor, Pressler. The normal form factor is the volume of the tree divided by the product of height and the sectional area at a height in fixed proportion to the height of the tree. This proportion is generally assumed as 1/20. Then Normal form i HXB (taken at =) 20 97. The Conception and Use of Form Exponents.—Analytical geometry teaches us the mathematical expression for the curves whose revolution about an axis will form a cone, an Appolonian paraboloid, a Neilian paraboloid, etc. 7 The straight line parallel to the axis is represented by the equation y= p:x. | * Sometimes called Rinniker’s absolute form factor. FORM FACTORS. 183 The curve generating an Appolonian paraboloid, by the equation y* = pow. The curve generating a cone, by the equation y? = p3x?. That generating a Neilian paraboloid by the equation yy? = pax’, in which x is the abscissa, y the ordinate, and ~, po, p3, pa are constants, the so-called parameters. The general expression applicable to any one of these curves is P= px", The form of the body produced depends on the exponent 7, which is called the form exponent or form coefficient. It is possible to determine the form exponent of a stem or log by a few measurements. Suppose that it is desired to find the exponent of form of a log such as that shown in Fig. 36. Let , be the distance yy, Y, ! | ee ten =e Fic. 36. from the tip of the tree to the base of the log, x2 the distance from the tip of the tree to the top of the log, y, the radius of the base, and 42 the radius of the top. Then 4? = px and V2" = px". If the first equation is divided by the second, then from which r(log +, —log x2) = 2(log 4 —log ya), 154 FOREST MENSURATION. log Wi — log Y2 =u9 -- : — log +, —log #2 Exam ple.—Suppose that the two measured diameters are 18 and 15 inches,* and the lengths are 61 and 45 feet. log y,=log 1.5 =0.176091 log x, =log 61 =1.785330 log ye=log 1.25 =0.086g910 log x2=log 45 =1.653213 Difference =0.089181 Difference =0.132117 o 5c 02089181 Bey Soe iar Ta 75 =1.35 The cubic contents of the stem of a tree may be determined by the formula in which V is the volume, D the diameter at the base, H the length and r the form exponent. In this formula the diameter is taken at the base of the tree. For the breast-height diameter the fol- lowing formula must be used: \ Space does not. permit a full mathematical explanation of the derivation of these formule. An algebraic derivation of the formula is described in Miiller’s Holzmesskunde, page 12, and a proof by calculus is contained in Lehr- und Hand- buch der Holzmesskunde, by Langenbacher und Nosseck, page 68. 98. Form Height.—By form height is meant the product of the form factor by the height of a given tree. The German tables of form heights are constructed for convenience in determining the cubic contents of trees by the use of form factors. —— ie esi 215") i loom * In practical calculations it is immaterial whether the diameters or radii are used. : FORM FACTORS, 185 99. Special Methods of Determining Form Factors.—Philipp’s Method.— In 1896 Karl Philipp of Baden published, in a pamphlet entitled Hilfstabellen fiir Forst-taxatoren, a new formula which serves as a mathematical expres- sion for all forms of the stems of trees. The equation representing any stem curve is as follows: yn m= xm —n. In using this formula, the form factor is based on the volume and height of the stem above breast-height (1.3 meters). The form factor is expressed n : ; by the fraction os Suppose that the form factor is 0.47, expressed as a vulgar n fraction ae then 2=47 and m=1oo, and the equation reads: y** it px*, The formula is used to determine the form factor of a tree in the follow- ing way: Let D be the breast-height diameter, d the diameter at a point on the stem representing a fixed and arbitrary proportion of the height (usually 0.4 of the height), H the height above breast-height, and h the distance from the point where d is measured to the tip. ‘hen = I D d. log - —log a 12 ioe H—log h 3 | Strzelecki’s Method.—The following method was described by Forst- director Heinrich Ritter von Strzelecki, a Galician forester, in the Central- blatt fiir das gesammte Forstwesen, 1883. It is interesting and may in some cases prove of practical value. With a paraboloid as a model, it is assumed that, if d is the diameter at one-half the height and D is the basal diameter, then d=V3D=0.707D, or <=0.707. d The quotient D is greater or less than 0.707, according as the trunk of the tree is greater or less than a paraboloid, and in the same proportion as the form factor will be greater or less than 0.50. Expressed in a formula, d - od 0.707 55 =0.5:F, or #=0.707 5, 186 FOREST MENSURATION. ‘The volume of the tree would then be V=o é